observational units were randomly assigned to the treatments - only systematic influence on the outcome should be due to the treatment - treatment effect
data("InsectSprays")
with(InsectSprays, tapply(count, spray, length))
## A B C D E F
## 12 12 12 12 12 12
with(InsectSprays, tapply(count, spray, summary))
## $A
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 7.00 11.50 14.00 14.50 17.75 23.00
##
## $B
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 7.00 12.50 16.50 15.33 17.50 21.00
##
## $C
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.000 1.500 2.083 3.000 7.000
##
## $D
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.000 3.750 5.000 4.917 5.000 12.000
##
## $E
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.00 2.75 3.00 3.50 5.00 6.00
##
## $F
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 9.00 12.50 15.00 16.67 22.50 26.00
with(InsectSprays, tapply(count, spray, sd))
## A B C D E F
## 4.719399 4.271115 1.975225 2.503028 1.732051 6.213378
stripplots for less than 15 points per treatment interpretation: higher or lower values for the dependent variable? variance?
library(ggplot2)
ggplot(InsectSprays, aes(spray, count))+geom_point(shape=1, position=position_jitter(width=0.1, height=0))
library(lattice)
stripplot(count~spray, data=InsectSprays, jitter=0.2)
boxplots Interpretation: median differences? variance? - compare box heights skewness? - whiskers and median outliers? - data points outside the whiskers
plot(count~spray, data=InsectSprays)
Another visualization: Both plots contain the same information the means +- their standard error is plotted
library(sciplot)
bargraph.CI(spray, count, col=(gray(0.88)), data=InsectSprays, xlab='spray', ylab='count', ylim=c(0,20))
lineplot.CI(spray, count, type='p', data=InsectSprays, xlab='spray', ylab='count', ylim=c(0,20))
InsectSprays$block=factor(rep(rep(1:6, each=2), times=6))
ggplot(InsectSprays, aes(spray, count)) + geom_point() + facet_grid(~block)
s.o <- with(InsectSprays, reorder(spray, count, mean, order = TRUE))
InsectSprays$spray.o <- factor(InsectSprays$spray, levels = levels(s.o))
b.o <- with(InsectSprays, reorder(block, count, mean, order = TRUE))
InsectSprays$block.o <- factor(InsectSprays$block, levels = levels(b.o))
ggplot(InsectSprays, aes(spray.o, count)) + geom_point() + facet_grid( ~ block.o) + xlab("spray")
visualization accounting for block structure the second plot orders the blocks and treatments by count means asses: do blocks themselves have an effect? do treatment effects depend on the blocks? here no block effects visible
Assumptions: independence, equal variance, normal distributed measurements for each treatment follow a normal distribution with mean \(\mu\) and variance \(\sigma\)2, assumed to be fixed unkown numbers with the variance being the same for each treatment.
ins.lm = lm(count~spray, data=InsectSprays)
anova(ins.lm)
#same results:
ins.aov = aov(count~spray, data=InsectSprays)
summary(ins.aov)
## Df Sum Sq Mean Sq F value Pr(>F)
## spray 5 2669 533.8 34.7 <2e-16 ***
## Residuals 66 1015 15.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
If the p-value of the overall F test is smaller than the significance level, reject the null hypothesis and conclude that not all the treatments have the same theoretical means. Only valid if model assumptions hold.
spray - Sum Sq: SSB, if SSB=0 (i.e. SST=SSW) all sample treatments means are equal to the overall mean. If the null hypothesis is true SSB should be small otherwise it should be big. spray - Mean Sq: MSB=SSB/(k-1) quantifies the variability between the treatment means. Residuals - Sum Sq: SSW, if SSW=0 (i.e. SST=SSB) within every treatment, all the values are equal to the respective mean.
residuals := observed values - fitted values RMSE := root mean squared error (estimates the standard deviation \(\sigma\) of our observations) R2 := 1-SSW/SST (the coefficient of determination: is the proportion of variance of the dependent variable explained by the model)
# RMSE
summary(ins.lm)$sigma
## [1] 3.921902
#R^2^
summary(ins.lm)$r.squared
## [1] 0.724439
ins.lm.noint = lm(count~spray-1, data=InsectSprays)
summary(ins.lm.noint)$coefficients
## Estimate Std. Error t value Pr(>|t|)
## sprayA 14.500000 1.132156 12.807428 1.470512e-19
## sprayB 15.333333 1.132156 13.543487 1.001994e-20
## sprayC 2.083333 1.132156 1.840148 7.024334e-02
## sprayD 4.916667 1.132156 4.342749 4.953047e-05
## sprayE 3.500000 1.132156 3.091448 2.916794e-03
## sprayF 16.666667 1.132156 14.721181 1.573471e-22
excludes intercept since we do not have a control treatment, shows two-sided t test wether the respective group mean is zero.
the multiple testing problem If we perform a family of hypothesis tests (example compare pairwise k treatments: m = k*(k-1)/2 hypothesis tests). The probability of committing at least one Type I error in all the tests together is called the family error rate (1-(1-\(\alpha\))m). The increase of the familywise error rate is called the multiple testing problem.
#with(InsectSprays, pairwise.t.test(count, spray, 'holm'))
#with(InsectSprays, pairwise.wilcox.test(count, spray, 'holm'))
Gives adjusted p-values so that the family error rate is 0.05. Commented here since it gives too many warnings..
Pairwise comparison of treatment means. assumptions: normality, roughly similar variances sensitive to outliers and conservative if not all treatments have the same number of observations.
TukeyHSD(ins.aov, 'spray')
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = count ~ spray, data = InsectSprays)
##
## $spray
## diff lwr upr p adj
## B-A 0.8333333 -3.866075 5.532742 0.9951810
## C-A -12.4166667 -17.116075 -7.717258 0.0000000
## D-A -9.5833333 -14.282742 -4.883925 0.0000014
## E-A -11.0000000 -15.699409 -6.300591 0.0000000
## F-A 2.1666667 -2.532742 6.866075 0.7542147
## C-B -13.2500000 -17.949409 -8.550591 0.0000000
## D-B -10.4166667 -15.116075 -5.717258 0.0000002
## E-B -11.8333333 -16.532742 -7.133925 0.0000000
## F-B 1.3333333 -3.366075 6.032742 0.9603075
## D-C 2.8333333 -1.866075 7.532742 0.4920707
## E-C 1.4166667 -3.282742 6.116075 0.9488669
## F-C 14.5833333 9.883925 19.282742 0.0000000
## E-D -1.4166667 -6.116075 3.282742 0.9488669
## F-D 11.7500000 7.050591 16.449409 0.0000000
## F-E 13.1666667 8.467258 17.866075 0.0000000
library(agricolae)
HSD.test(ins.lm, 'spray', group=TRUE, console=TRUE)
##
## Study: ins.lm ~ "spray"
##
## HSD Test for count
##
## Mean Square Error: 15.38131
##
## spray, means
##
## count std r Min Max
## A 14.500000 4.719399 12 7 23
## B 15.333333 4.271115 12 7 21
## C 2.083333 1.975225 12 0 7
## D 4.916667 2.503028 12 2 12
## E 3.500000 1.732051 12 1 6
## F 16.666667 6.213378 12 9 26
##
## Alpha: 0.05 ; DF Error: 66
## Critical Value of Studentized Range: 4.150851
##
## Minimun Significant Difference: 4.699409
##
## Treatments with the same letter are not significantly different.
##
## count groups
## F 16.666667 a
## B 15.333333 a
## A 14.500000 a
## D 4.916667 b
## E 3.500000 b
## C 2.083333 b
either values with p adj < \(\alpha\) or different letters are significantly different
We want to test if the treatment mean is different from the control treatment mean
data("InsectSprays")
ins.lm = lm(count~spray, data=InsectSprays)
summary(ins.lm)$coefficient
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.5000000 1.132156 12.8074279 1.470512e-19
## sprayB 0.8333333 1.601110 0.5204724 6.044761e-01
## sprayC -12.4166667 1.601110 -7.7550382 7.266893e-11
## sprayD -9.5833333 1.601110 -5.9854322 9.816910e-08
## sprayE -11.0000000 1.601110 -6.8702352 2.753922e-09
## sprayF 2.1666667 1.601110 1.3532281 1.805998e-01
contrasts(InsectSprays$spray)
## B C D E F
## A 0 0 0 0 0
## B 1 0 0 0 0
## C 0 1 0 0 0
## D 0 0 1 0 0
## E 0 0 0 1 0
## F 0 0 0 0 1
with(InsectSprays, tapply(count, spray, mean))
## A B C D E F
## 14.500000 15.333333 2.083333 4.916667 3.500000 16.666667
the intercept estimates the mean of the reference level while all other coefficients estimate differences of the respective treatment to the mean of the reference treatment. Estimated mean insects surviving a treatment with reference level (spray A) is 14.5. For the reference level the t test measures if the mean is zero, for the other treatments it measures if the treatment mean is different from the reference mean.
options('contrasts')
## $contrasts
## unordered ordered
## "contr.treatment" "contr.poly"
Sometimes we want to know if the treatment differs from the grand mean, the average of all treatment means.
ins.lm.sum = lm(count~spray, data=InsectSprays, contrast=list(spray='contr.sum'))
summary(ins.lm.sum)$coefficients
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.500000 0.4622006 20.553848 2.160853e-30
## spray1 5.000000 1.0335119 4.837874 8.223823e-06
## spray2 5.833333 1.0335119 5.644186 3.778197e-07
## spray3 -7.416667 1.0335119 -7.176180 7.866853e-10
## spray4 -4.583333 1.0335119 -4.434718 3.571783e-05
## spray5 -6.000000 1.0335119 -5.805449 2.003785e-07
InsectSprays$spray.sum = InsectSprays$spray
contrasts(InsectSprays$spray.sum) = 'contr.sum'
contrasts(InsectSprays$spray.sum)
## [,1] [,2] [,3] [,4] [,5]
## A 1 0 0 0 0
## B 0 1 0 0 0
## C 0 0 1 0 0
## D 0 0 0 1 0
## E 0 0 0 0 1
## F -1 -1 -1 -1 -1
The intercept is now an estimate for the overall mean. The contrast matrix shows that spray F is now interpreted as the reference level.
Reducing the number of comparisons is attractive since it makes the p value correction less strict, such that we gain in power.
The multcomp library includes:
library(multcomp)
## Loading required package: mvtnorm
## Loading required package: survival
## Loading required package: TH.data
## Loading required package: MASS
##
## Attaching package: 'TH.data'
## The following object is masked from 'package:MASS':
##
## geyser
summary(glht(ins.lm, mcp(spray='Dunnett')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Dunnett Contrasts
##
##
## Fit: lm(formula = count ~ spray, data = InsectSprays)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## B - A == 0 0.8333 1.6011 0.520 0.979
## C - A == 0 -12.4167 1.6011 -7.755 <1e-04 ***
## D - A == 0 -9.5833 1.6011 -5.985 <1e-04 ***
## E - A == 0 -11.0000 1.6011 -6.870 <1e-04 ***
## F - A == 0 2.1667 1.6011 1.353 0.526
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
p values are adjusted. Reference level is A.
Test specific hypotheses: pass contrast matrix Here we compare the means of treatment spray B and treatment spray F.
K = matrix(c(0,1,0,0,0,-1), nrow=1)
summary(glht(ins.lm, linfct=K))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Fit: lm(formula = count ~ spray, data = InsectSprays)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## 1 == 0 -1.333 1.601 -0.833 0.408
## (Adjusted p values reported -- single-step method)
summary(glht(ins.lm, mcp(spray = 'GrandMean')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: GrandMean Contrasts
##
##
## Fit: lm(formula = count ~ spray, data = InsectSprays)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## A == 0 5.000 1.034 4.838 < 1e-04 ***
## B == 0 5.833 1.034 5.644 < 1e-04 ***
## C == 0 -7.417 1.034 -7.176 < 1e-04 ***
## D == 0 -4.583 1.034 -4.435 0.000218 ***
## E == 0 -6.000 1.034 -5.805 < 1e-04 ***
## F == 0 7.167 1.034 6.934 < 1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
Check if the residuals come from a normal distribution by comparing the quantiles of the residual distribution with the quantiles of a standard normal distribution. Do not use for less than 30 samples.
library(car)
## Loading required package: carData
qqPlot(resid(ins.lm))
## [1] 69 70
Reject the null hypothesis of normality if p < \(\alpha\)
shapiro.test(resid(ins.lm))
##
## Shapiro-Wilk normality test
##
## data: resid(ins.lm)
## W = 0.96006, p-value = 0.02226
For small samples a nonsignificant Shapiro Wilk test does not imply no problems with normality are present.
plot(fitted(ins.lm), resid(ins.lm), las=1, xlab='Fitted Values', ylab='Residuals')
abline(h=0)
plot(ins.lm, which=1)
The variance of the residuals seems to increase with the fitted values, we conclude that the assumption of equal variance is violkated.
high variance implies a low reliability
bartlett.test(count~spray, data=InsectSprays)
##
## Bartlett test of homogeneity of variances
##
## data: count by spray
## Bartlett's K-squared = 25.96, df = 5, p-value = 9.085e-05
leveneTest(count~spray, data=InsectSprays)
if p < \(\alpha\) reject the null hypothesis of equal variance.
oneway.test(count~spray, data=InsectSprays)
##
## One-way analysis of means (not assuming equal variances)
##
## data: count and spray
## F = 36.065, num df = 5.000, denom df = 30.043, p-value = 7.999e-12
Violations of the model assumption can very severely inflate the probability of a Type I error, i.e. that the F test rejects the true null hypothesis of no treatment effects too often on average) and diminish the power, i.e. that the F test too often fails to reject false null hypothesis.
Assumptions:
The parametric model is retrieved if one assumes that F is a normal distribution function
Tests the null hypothesis that each of the underlying distributions is the same against the alternative that at least two treatment effects differ. If the null hypothesis is true mean ranks should not differ too much between treatments - has problems with ties! Suffers if the variances are not equal.
kruskal.test(count~spray, data=InsectSprays)
##
## Kruskal-Wallis rank sum test
##
## data: count by spray
## Kruskal-Wallis chi-squared = 54.691, df = 5, p-value = 1.511e-10
To test which treatments differ from each other, we apply a special ad hoc procedures to perform all two-sided pairwise comparisons.
library(NSM3)
## Loading required package: combinat
##
## Attaching package: 'combinat'
## The following object is masked from 'package:utils':
##
## combn
## Loading required package: partitions
##
## Attaching package: 'partitions'
## The following object is masked from 'package:car':
##
## S
#with(InsectSprays, pSDCFlig(x=count, g=as.numeric(spray), method=NA))
Not affected as much if we do not have equal variance. Tests the nullhypotheis that the distributions of the numeric variable are the same in each treatment.
library(asbio)
## Loading required package: tcltk
## Registered S3 method overwritten by 'asbio':
## method from
## print.ci coin
with(InsectSprays, BDM.test(count,spray))
##
## One way Brunner-Dette-Munk test
##
## df1 df2 F* P(F > F*)
## 4.828351 63.19189 44.26642 4.138278e-19
useful for data with outliers and tests the null hypothesis of equal trimmed means instead of medians.
library(asbio)
with(InsectSprays, trim.test(count, spray, tr=0.2))
## $Results
## df1 df2 F* P(>F)
## 1 5 18.92297 28.22811 3.745716e-08
library(lmPerm)
anova(lmp(count~spray, data=InsectSprays))
## [1] "Settings: unique SS "
## Analysis of Variance Table
##
## Response: count
## Df R Sum Sq R Mean Sq Iter Pr(Prob)
## spray 5 2668.8 533.77 5000 < 2.2e-16 ***
## Residuals 66 1015.2 15.38
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Blocking: grouping of homogeneous units to remove block-to-block variation from the experimental error. 1. Natural discrete blocks: The plots may naturally be grouped in blocks. example: piglets from the same mother
Continuous gradients The plots may show differences which seem to change along a continous gradient. This is often due to plot homogeneity in space or time. example: agricultural fields
Trial management blocks Many experiments need more staff than one person, trials might be executed differently. If possible staff assignment should reflect staff assignment.
Principles for blocking: 1. all blocks should have the same size 2. be sufficiently big to apply each treatment at least once per block
In the randomized complete block design (RCBD), every treatment is applied in every plot exactly one time, allocating treatments randomly within blocks.
In the generalized complete block design (GRCBD) every treatment us applied in every block exactly r > 1 times, allocating treatments randomly within blocks.
Only takes the fixed effects of the blocks and the treatments into account, treatment effects are the same in each block. There is no block x treatment interaction. Usually not what we want: 1. The interest is usually not in the fixed effect of the block 2. estimation of the block effects takes too many degrees of freedom 3. we do not care if the block effect is significant, we only want to correctly estimate the treatment effect while accounting for the plot structure induced by the blocks. 4. The assumption that the error terms are independent might not be particularly realistic.
Accounts for the random block effects. The main interest is not in the effect of the blocks, only in the variation introduced by the blocking factor. We have a fixed treatment effect and two random components the block effect and the error terms, both normally distributed with mean zero.
Do not ignore block effects! Otherwise the variability due to block effects falls into the errors standard deviation making it much harder to detect the real treatment effect!
The data set must have one variable for the blocks - long format
InsectSprays$block = factor(rep(rep(1:6, each=2), times=6))
head(InsectSprays)
library(nlme)
ins.lme = lme(sqrt(count)~spray, random=~1|block, data=InsectSprays)
One random effect, a random intercept per block. For the InsectSprays data we use a square root transformation for the data. The REML (restricted maximum likelihood) is the default of lme, and is used to get better variance estimates.
summary(ins.lme)
## Linear mixed-effects model fit by REML
## Data: InsectSprays
## AIC BIC logLik
## 152.2289 169.7461 -68.11445
##
## Random effects:
## Formula: ~1 | block
## (Intercept) Residual
## StdDev: 0.2540835 0.579766
##
## Fixed effects: sqrt(count) ~ spray
## Value Std.Error DF t-value p-value
## (Intercept) 3.760678 0.1969021 61 19.099225 0.0000
## sprayB 0.115953 0.2366885 61 0.489897 0.6260
## sprayC -2.515822 0.2366885 61 -10.629254 0.0000
## sprayD -1.596325 0.2366885 61 -6.744412 0.0000
## sprayE -1.951217 0.2366885 61 -8.243821 0.0000
## sprayF 0.257939 0.2366885 61 1.089782 0.2801
## Correlation:
## (Intr) sprayB sprayC sprayD sprayE
## sprayB -0.601
## sprayC -0.601 0.500
## sprayD -0.601 0.500 0.500
## sprayE -0.601 0.500 0.500 0.500
## sprayF -0.601 0.500 0.500 0.500 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.4815817 -0.5213930 -0.1471345 0.6234485 1.9749960
##
## Number of Observations: 72
## Number of Groups: 6
fixef(ins.lme)
## (Intercept) sprayB sprayC sprayD sprayE sprayF
## 3.7606784 0.1159530 -2.5158217 -1.5963245 -1.9512174 0.2579388
Gives: 1. standard deviation of the blocks and the residuals, shows if the block effect is negligible compared to the errors. 2. AIC and BIC criterion, smaller values are preferable 3. REML estimates of the foxed effects with the usual t tests and the correlations of the fixed effects estimator.
The random effects are not estimated but predicted, one for each block.
ranef(ins.lme)
ranef(ins.lme)$'(Intercept)'
## [1] -0.30503308 0.28296568 -0.01645621 -0.10988229 0.19387989 -0.04547399
The random effects show how a particular block compares to a ‘typical’ block with random effect of zero. Negative values mean that the block has less insects than a typical block, positive values mean higher counts.
Compare block sample means to the overall mean
blk.centred = with(InsectSprays, tapply(sqrt(count), block, mean)-mean(sqrt(count)))
blk.ranef = ranef(ins.lme)$'(Intercept)'
cbind(blk.centred, blk.ranef, ratio = blk.centred/blk.ranef)
## blk.centred blk.ranef ratio
## 1 -0.43738127 -0.30503308 1.433881
## 2 0.40573924 0.28296568 1.433881
## 3 -0.02359626 -0.01645621 1.433881
## 4 -0.15755818 -0.10988229 1.433881
## 5 0.27800078 0.19387989 1.433881
## 6 -0.06520431 -0.04547399 1.433881
All the random effects predictions are essentially the deviations of the block means from the overall mean, but shrunk towards zero by the same factor.
within group fitted values include random effects or level to zero to only include fixed effects
head(fitted(ins.lme))
## 1 1 2 2 3 3
## 3.455645 3.455645 4.043644 4.043644 3.744222 3.744222
head(fitted(ins.lme, level=0))
## 1 1 2 2 3 3
## 3.760678 3.760678 3.760678 3.760678 3.760678 3.760678
The same applies to residuals.
Equal variance
plot(ins.lme)
Does the variance of the error terms change with the fitted values? Wedge shape? The lme function can handle modelled different variances, for example different variances according to the levels of the factor.
Normality
qqPlot(resid(ins.lme))
## 5 6
## 34 23
shapiro.test(resid(ins.lme))
##
## Shapiro-Wilk normality test
##
## data: resid(ins.lme)
## W = 0.98985, p-value = 0.8351
shapiro.test(ranef(ins.lme)$'(Intercept)')
##
## Shapiro-Wilk normality test
##
## data: ranef(ins.lme)$"(Intercept)"
## W = 0.96421, p-value = 0.8515
Typically testing for normality of the random effect of the blocks does not make sense since we have less than 30 blocks and the tests are not meaningful.
Give a first impression about the precision of the estimators, but are sensitiv to outliers and ill-behaved data sets (nonnormality, unequal variances)
intervals(ins.lme)
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 3.3669482 3.7606784 4.1544086
## sprayB -0.3573348 0.1159530 0.5892408
## sprayC -2.9891095 -2.5158217 -2.0425339
## sprayD -2.0696124 -1.5963245 -1.1230367
## sprayE -2.4245052 -1.9512174 -1.4779296
## sprayF -0.2153491 0.2579388 0.7312266
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: block
## lower est. upper
## sd((Intercept)) 0.1041355 0.2540835 0.6199464
##
## Within-group standard error:
## lower est. upper
## 0.4855018 0.5797660 0.6923322
anova(ins.lme, type='marginal')
Test if all fxed effects estimates except the intercept are zero. If the null hypothesis is rejected we get a significant effect of the treatments (we do not know which one).
We can set contrasts and use the multcomp library to make pairwise comparisons.
library(multcomp)
summary(glht(ins.lme, mcp(spray='GrandMean')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: GrandMean Contrasts
##
##
## Fit: lme.formula(fixed = sqrt(count) ~ spray, data = InsectSprays,
## random = ~1 | block)
##
## Linear Hypotheses:
## Estimate Std. Error z value Pr(>|z|)
## A == 0 0.9482 0.1528 6.207 < 1e-04 ***
## B == 0 1.0642 0.1528 6.965 < 1e-04 ***
## C == 0 -1.5676 0.1528 -10.260 < 1e-04 ***
## D == 0 -0.6481 0.1528 -4.242 0.00017 ***
## E == 0 -1.0030 0.1528 -6.565 < 1e-04 ***
## F == 0 1.2062 0.1528 7.895 < 1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
summary(glht(ins.lme, mcp(spray='Tukey')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Tukey Contrasts
##
##
## Fit: lme.formula(fixed = sqrt(count) ~ spray, data = InsectSprays,
## random = ~1 | block)
##
## Linear Hypotheses:
## Estimate Std. Error z value Pr(>|z|)
## B - A == 0 0.1160 0.2367 0.490 0.99654
## C - A == 0 -2.5158 0.2367 -10.629 < 0.001 ***
## D - A == 0 -1.5963 0.2367 -6.744 < 0.001 ***
## E - A == 0 -1.9512 0.2367 -8.244 < 0.001 ***
## F - A == 0 0.2579 0.2367 1.090 0.88568
## C - B == 0 -2.6318 0.2367 -11.119 < 0.001 ***
## D - B == 0 -1.7123 0.2367 -7.234 < 0.001 ***
## E - B == 0 -2.0672 0.2367 -8.734 < 0.001 ***
## F - B == 0 0.1420 0.2367 0.600 0.99107
## D - C == 0 0.9195 0.2367 3.885 0.00152 **
## E - C == 0 0.5646 0.2367 2.385 0.16114
## F - C == 0 2.7738 0.2367 11.719 < 0.001 ***
## E - D == 0 -0.3549 0.2367 -1.499 0.66468
## F - D == 0 1.8543 0.2367 7.834 < 0.001 ***
## F - E == 0 2.2092 0.2367 9.334 < 0.001 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
Used for comparing treatments especially in more complex settings estimated marginal means: The idea is to define a reference grid of interesting values (for factors these are just their levels) and then compute the fitted values according to the mdoel on the reference grid.
library(emmeans)
ref_grid(ins.lme)
## 'emmGrid' object with variables:
## spray = A, B, C, D, E, F
## Transformation: "sqrt"
ins.emm = emmeans(ins.lme,'spray')
ins.emm
## spray emmean SE df lower.CL upper.CL
## A 3.76 0.197 5 3.255 4.27
## B 3.88 0.197 5 3.370 4.38
## C 1.24 0.197 5 0.739 1.75
## D 2.16 0.197 5 1.658 2.67
## E 1.81 0.197 5 1.303 2.32
## F 4.02 0.197 5 3.512 4.52
##
## Degrees-of-freedom method: containment
## Results are given on the sqrt (not the response) scale.
## Confidence level used: 0.95
pairs(ins.emm)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
## contrast estimate SE df t.ratio p.value
## A - B -0.116 0.237 61 -0.490 0.9964
## A - C 2.516 0.237 61 10.629 <.0001
## A - D 1.596 0.237 61 6.744 <.0001
## A - E 1.951 0.237 61 8.244 <.0001
## A - F -0.258 0.237 61 -1.090 0.8836
## B - C 2.632 0.237 61 11.119 <.0001
## B - D 1.712 0.237 61 7.234 <.0001
## B - E 2.067 0.237 61 8.734 <.0001
## B - F -0.142 0.237 61 -0.600 0.9907
## C - D -0.919 0.237 61 -3.885 0.0033
## C - E -0.565 0.237 61 -2.385 0.1776
## C - F -2.774 0.237 61 -11.719 <.0001
## D - E 0.355 0.237 61 1.499 0.6659
## D - F -1.854 0.237 61 -7.834 <.0001
## E - F -2.209 0.237 61 -9.334 <.0001
##
## Note: contrasts are still on the sqrt scale
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 6 estimates
plot(ins.emm, comparison=TRUE)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
The blue bars of the plots depicts confidence intervals of the marginal means. The red arrows are for the pairwise comparisons among the spray means. If any arrow from one spray overlaps with an arrow from another spray, their mean difference is not significant.
If we do not account for the block structure the estimate for of the errors standard deviation is bigger in the model without block effects, since the variability of the blocks is not properly accounted for.
Observations from different blocks are independent according to the definition of a mixed model. Observations from the same block are correlated because they share the block random effect. All observations in the same block share the same correlation given by the intraclass correlation coefficient. (block variance over sum block variance and error variance)
V = VarCorr(ins.lme)
V = as.numeric(V)
V[1]/(V[1]+V[2])
## [1] 0.1611194
Due to the definition used here negative correlations are not possible and can only be attained if the model is defined in a different way. Negative correlation can be expected if there is competition inside blocks for example for food and light.
Because we have to account for the random effect of the blocks the decomposition of the sum of squares no longer works as it does in linear models. As a result no values for R2 are generally given. If we want to quantify the fit of the model we can give the marginal and conditional R2 values extracted as follows.
library(MuMIn)
r.squaredGLMM(ins.lme)
## Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.
## R2m R2c
## [1,] 0.7566123 0.7958267
The immer data contains the yield of five varieties (M, P, S, T and V) of barley grown in six locations Loc (C, D, GR, M, UF and W) for two years (Y1 and Y2).
library(MASS)
data(immer)
immer
ggplot(immer, aes(x=Var, y=Y1)) + geom_point() + facet_grid(~Loc)
The data inside the locations is quite homogeneous but the locations greatly differ from each other. If we ignoring the locations will result in pooling heterogeneous data together. This will be reflected in the residual variance which will be overestimated and we do not get a significant treatment (variety) effect. If we inlcude the location (blocks) as a fixed effect we do not account for the randomization which happens inside locations. Since we are not interested in the fixed effect but rather in the random effect introduced trough the location we can treat it as a blocking factor. (Only include locations as a fixed effect if you are excactly interested in those locations.)
immer.lme = lme(Y1~Var, random=~1|Loc, data=immer)
immer.lme
## Linear mixed-effects model fit by REML
## Data: immer
## Log-restricted-likelihood: -111.3314
## Fixed: Y1 ~ Var
## (Intercept) VarP VarS VarT VarV
## 102.5833333 7.1666667 -0.5500000 24.8166667 0.8833333
##
## Random effects:
## Formula: ~1 | Loc
## (Intercept) Residual
## StdDev: 26.08863 12.76273
##
## Number of Observations: 30
## Number of Groups: 6
anova(immer.lme)
We have a significant location effect. The standard deviation of the random intercepts is even bigger than the residual standard deviation, because the blocks are so different. The only drawback of this analysis is that the sample is so small and it is very difficult asses the model assumptions in a reliable way. We could use a non parametric analysis instead.
plot(immer.lme)
qqPlot(resid(immer.lme))
## UF M
## 3 14
shapiro.test(resid(immer.lme))
##
## Shapiro-Wilk normality test
##
## data: resid(immer.lme)
## W = 0.96611, p-value = 0.4389
The Friedman’s test is a non parametric test of the null hypothesis that the treatment effect is the same for every level of the treatment in RCBD. The test idea is to rank observations within blocks and then sum ranks for the treatments over all the blocks. The ranking makes the method robust to outliers - this robustness is paid for some loss of power (compared to the parametric model) if the data do come from a normal distribution. Assumptions: the distribution of the dependent variables for the levels of the treatment are shifted variants of the same distribution. The test should not be used for widely different distributions or if the treatments work differently for some block, i.e. if we have treatment x block interactions.
friedman.test(Y1~Var|Loc, immer)
##
## Friedman rank sum test
##
## data: Y1 and Var and Loc
## Friedman chi-squared = 10.933, df = 4, p-value = 0.02732
We get a significant variety effect.
If we have GRCBD (more than one observation per treatment per block) we can answer the question whether the treatment effects are different by block. Visualization by plotting means for each combination of block and treatments interaction plot
with(InsectSprays, interaction.plot(spray, block, sqrt(count)))
No interaction: the traces should be more or less parallel. Approach: fit a model with nested random effects. The blocks are a random sample from the set of all possible blocks, so that any differences in treatment effects due to the blocks should also be treated as random effects. We can fit a model with two levels of random effects: one level for the blocks and one level for the treatments within each block.
ins.lme.x = lme(sqrt(count)~spray, random=~1|block/spray, data=InsectSprays)
ins.lme.x
## Linear mixed-effects model fit by REML
## Data: InsectSprays
## Log-restricted-likelihood: -67.43423
## Fixed: sqrt(count) ~ spray
## (Intercept) sprayB sprayC sprayD sprayE sprayF
## 3.7606784 0.1159530 -2.5158217 -1.5963245 -1.9512174 0.2579388
##
## Random effects:
## Formula: ~1 | block
## (Intercept)
## StdDev: 0.2394903
##
## Formula: ~1 | spray %in% block
## (Intercept) Residual
## StdDev: 0.2706021 0.5254596
##
## Number of Observations: 72
## Number of Groups:
## block spray %in% block
## 6 36
The log-likelihood of the more complex model is slightly higher, is the use of the more complex model justified? Since we fitted the model with the maximum likelihood method we use the likelihood-ratio test to quantify whether the improvement of the likelihood is sufficient given the additional model complexity.
anova(ins.lme, ins.lme.x)
Give the simpler model first. The simpler model seems to be sufficient, i.e. we do not have any evidence for an interaction of treatment effects and blocks. Thus we should proceed with the simpler model.
The turnip yield data The question is how the planting density in kg/ha affects the mean yield of turnips of two genotypes. Turn planting density in a factor first.
library(agridat)
turnip = mcconway.turnip
turnip$density = as.factor(turnip$density)
ggplot(turnip, aes(x = density, y = yield)) + geom_point() + facet_grid(~gen)
head(turnip)
tail(turnip)
with(turnip, tapply(yield, list(gen, density), mean))
## 1 2 4 8
## Barkant 2.9375 4.4 9.0875 9.6625
## Marco 1.3375 2.3 5.5625 7.7250
Visualization and means of every combination of density and genotype.
Questions: 1. Is there a planting density effect? - look at the visualization 2. Is there a genotype effect vary among the two genotype?
qplot(density, yield, data=turnip, facets=~block, shape=gen, size=I(2))
Block differences are clearly visible, Block 4 seems to have really low yield, and we can see an overall increase of the yield with the planting density.
Interaction plot
ggplot(turnip, aes(x=density, linetype=gen, group=gen, y=yield)) + stat_summary(fun.y=mean, geom='point') + stat_summary(fun.y=mean, geom='line')
## Warning: `fun.y` is deprecated. Use `fun` instead.
## Warning: `fun.y` is deprecated. Use `fun` instead.
Do the lines look more or less parallel? Here they do.. no interaction effect
We want to study the influence of two factors on a numeric variable. We are interested in the fixed effects of both factors. Assumption: data come from a normal distribution with an equal variance for each group.
In comparison to the cell means model which does only account for finding effects of the two factors combined, we can use the factor effect model two find separate main effects of the two factors and the interaction effect of them.
turnip.full = lm(yield~gen*density, data=turnip)
coef(summary(turnip.full))
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.9375 1.522377 1.9295486 0.058734960
## genMarco -1.6000 2.152966 -0.7431609 0.460490758
## density2 1.4625 2.152966 0.6792955 0.499748857
## density4 6.1500 2.152966 2.8565245 0.005999627
## density8 6.7250 2.152966 3.1235980 0.002827723
## genMarco:density2 -0.5000 3.044754 -0.1642169 0.870151687
## genMarco:density4 -1.9250 3.044754 -0.6322351 0.529806221
## genMarco:density8 -0.3375 3.044754 -0.1108464 0.912134467
R automatically chooses a reference level. The intercept is the estimated yield for the Barkant (reference level) and density 1. The yield at density 1 for the Marco variety is estimated as (2.9375-1.6000). If the interaction effect is bigger than zero, than the combined effect of one level of factor and one level of factor Z is bigger than the sum of the main effects; this is called a positive interaction. If the interaction effect is smaller than zero, this is called a negative interaction. For a balanced design, the combination of the above estimates yield exactly the sample means.
with(turnip, tapply(yield, list(gen, density), mean))
## 1 2 4 8
## Barkant 2.9375 4.4 9.0875 9.6625
## Marco 1.3375 2.3 5.5625 7.7250
Is the model using two factors and their interaction effect any better than predicting the overall mean for every observation (null model)?
summary(turnip.full)
##
## Call:
## lm(formula = yield ~ gen * density, data = turnip)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2875 -2.5500 -0.2187 1.8156 14.9125
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.9375 1.5224 1.930 0.05873 .
## genMarco -1.6000 2.1530 -0.743 0.46049
## density2 1.4625 2.1530 0.679 0.49975
## density4 6.1500 2.1530 2.857 0.00600 **
## density8 6.7250 2.1530 3.124 0.00283 **
## genMarco:density2 -0.5000 3.0448 -0.164 0.87015
## genMarco:density4 -1.9250 3.0448 -0.632 0.52981
## genMarco:density8 -0.3375 3.0448 -0.111 0.91213
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.306 on 56 degrees of freedom
## Multiple R-squared: 0.3516, Adjusted R-squared: 0.2705
## F-statistic: 4.338 on 7 and 56 DF, p-value: 0.0006739
The overall F test is in the last line and is significant, one should not proceed with partial F tests if the overall F test is not significant.
Do we need the interaction of the two factors? Test the null hypothesis that all interaction effects are zero. Do we need a main factor? Test the null hypothesis that the coefficients of the main effect are zero.
anova(turnip.full)
First check if the interaction effect is significant only remove non-significant interaction effects. If the interaction effect is significant, do not remove the involved main effects. If the interaction effect is non-significant one can remove main effects if they are also non-significant to simplify the model.
A model with only the main effects and not the interaction effects
turnip.main = lm(yield~gen + density, data=turnip)
summary(turnip.full)
##
## Call:
## lm(formula = yield ~ gen * density, data = turnip)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2875 -2.5500 -0.2187 1.8156 14.9125
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.9375 1.5224 1.930 0.05873 .
## genMarco -1.6000 2.1530 -0.743 0.46049
## density2 1.4625 2.1530 0.679 0.49975
## density4 6.1500 2.1530 2.857 0.00600 **
## density8 6.7250 2.1530 3.124 0.00283 **
## genMarco:density2 -0.5000 3.0448 -0.164 0.87015
## genMarco:density4 -1.9250 3.0448 -0.632 0.52981
## genMarco:density8 -0.3375 3.0448 -0.111 0.91213
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.306 on 56 degrees of freedom
## Multiple R-squared: 0.3516, Adjusted R-squared: 0.2705
## F-statistic: 4.338 on 7 and 56 DF, p-value: 0.0006739
dat = expand.grid(gen=levels(turnip$gen), density=levels(turnip$density))
pred = predict(object=turnip.sqrt, newdata=dat, interval='confidence')
ci = cbind(dat, pred^2)
ci
Suppose we want to compare the yield of the two genotypes. You could now average over the planting densities, which should only be done if the genotype effect is more or less the same for each planting density (check interaction plot and interaction effect).
emmeans(turnip.sqrt, pairwise~gen)
## NOTE: Results may be misleading due to involvement in interactions
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
## $emmeans
## gen emmean SE df lower.CL upper.CL
## Barkant 2.34 0.144 56 2.05 2.63
## Marco 1.86 0.144 56 1.57 2.15
##
## Results are averaged over the levels of: density
## Results are given on the sqrt (not the response) scale.
## Confidence level used: 0.95
##
## $contrasts
## contrast estimate SE df t.ratio p.value
## Barkant - Marco 0.479 0.203 56 2.358 0.0219
##
## Results are averaged over the levels of: density
## Note: contrasts are still on the sqrt scale
On average, over all the planting densities, the Barnkant genotype produces significantly higher yields. If we do have interactions it would be better to compare the genotypes separately for each planting density.
turnip.sqrt.gen = emmeans(turnip.sqrt, pairwise~gen|density)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
turnip.sqrt.gen$contrasts
## density = 1:
## contrast estimate SE df t.ratio p.value
## Barkant - Marco 0.490 0.407 56 1.204 0.2336
##
## density = 2:
## contrast estimate SE df t.ratio p.value
## Barkant - Marco 0.557 0.407 56 1.371 0.1759
##
## density = 4:
## contrast estimate SE df t.ratio p.value
## Barkant - Marco 0.539 0.407 56 1.325 0.1906
##
## density = 8:
## contrast estimate SE df t.ratio p.value
## Barkant - Marco 0.332 0.407 56 0.817 0.4176
##
## Note: contrasts are still on the sqrt scale
Because the standard errors are now higher, the separate comparison no longer produce significant differences. This is why you should exclude non-significant interactions from a model. The model should not be more complex than it should have to be. We could also compare all the combinations of genotype and planting density pairwisely:
emmeans(turnip.sqrt, pairwise~gen+density)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
## $emmeans
## gen density emmean SE df lower.CL upper.CL
## Barkant 1 1.61 0.288 56 1.038 2.19
## Marco 1 1.12 0.288 56 0.548 1.70
## Barkant 2 2.02 0.288 56 1.448 2.60
## Marco 2 1.47 0.288 56 0.890 2.04
## Barkant 4 2.74 0.288 56 2.168 3.32
## Marco 4 2.20 0.288 56 1.629 2.78
## Barkant 8 2.98 0.288 56 2.403 3.55
## Marco 8 2.65 0.288 56 2.071 3.22
##
## Results are given on the sqrt (not the response) scale.
## Confidence level used: 0.95
##
## $contrasts
## contrast estimate SE df t.ratio p.value
## Barkant 1 - Marco 1 0.4896 0.407 56 1.204 0.9273
## Barkant 1 - Barkant 2 -0.4099 0.407 56 -1.008 0.9713
## Barkant 1 - Marco 2 0.1475 0.407 56 0.363 1.0000
## Barkant 1 - Barkant 4 -1.1297 0.407 56 -2.778 0.1214
## Barkant 1 - Marco 4 -0.5910 0.407 56 -1.453 0.8280
## Barkant 1 - Barkant 8 -1.3649 0.407 56 -3.357 0.0288
## Barkant 1 - Marco 8 -1.0329 0.407 56 -2.540 0.2005
## Marco 1 - Barkant 2 -0.8995 0.407 56 -2.212 0.3605
## Marco 1 - Marco 2 -0.3421 0.407 56 -0.841 0.9898
## Marco 1 - Barkant 4 -1.6194 0.407 56 -3.982 0.0046
## Marco 1 - Marco 4 -1.0806 0.407 56 -2.658 0.1577
## Marco 1 - Barkant 8 -1.8545 0.407 56 -4.561 0.0007
## Marco 1 - Marco 8 -1.5225 0.407 56 -3.744 0.0095
## Barkant 2 - Marco 2 0.5574 0.407 56 1.371 0.8664
## Barkant 2 - Barkant 4 -0.7198 0.407 56 -1.770 0.6421
## Barkant 2 - Marco 4 -0.1811 0.407 56 -0.445 0.9998
## Barkant 2 - Barkant 8 -0.9550 0.407 56 -2.349 0.2867
## Barkant 2 - Marco 8 -0.6230 0.407 56 -1.532 0.7869
## Marco 2 - Barkant 4 -1.2772 0.407 56 -3.141 0.0509
## Marco 2 - Marco 4 -0.7385 0.407 56 -1.816 0.6120
## Marco 2 - Barkant 8 -1.5124 0.407 56 -3.719 0.0103
## Marco 2 - Marco 8 -1.1804 0.407 56 -2.903 0.0913
## Barkant 4 - Marco 4 0.5387 0.407 56 1.325 0.8855
## Barkant 4 - Barkant 8 -0.2352 0.407 56 -0.578 0.9990
## Barkant 4 - Marco 8 0.0969 0.407 56 0.238 1.0000
## Marco 4 - Barkant 8 -0.7739 0.407 56 -1.903 0.5546
## Marco 4 - Marco 8 -0.4419 0.407 56 -1.087 0.9570
## Barkant 8 - Marco 8 0.3320 0.407 56 0.817 0.9915
##
## Note: contrasts are still on the sqrt scale
## P value adjustment: tukey method for comparing a family of 8 estimates
Problem: the sums of squares no longer add up as they do in the balanced case. Consequence: Distinguish between sequential and marginal F tests.
Marginal F tests asses whether an effect significantly reduces the resdiual sum of squares adjusted for all the other effects at the same or lower level. The order of terms in the model equation does not matter.
library(car)
Anova(turnip.sqrt, type=2)
How to account for the blocks of the turnip data? Use a linear mixed-effect model with random intercept per block:
turnip.lme = lme(sqrt(yield)~gen*density, random=~1|block, data=turnip)
This model combines the factorial treatment structure with the random intercept per block.
plot(turnip.lme)
qqPlot(resid(turnip.lme))
## B1 B4
## 25 12
shapiro.test(resid(turnip.lme))
##
## Shapiro-Wilk normality test
##
## data: resid(turnip.lme)
## W = 0.98437, p-value = 0.595
Normaility looks good, but the variance still increases with the fitted values.
Heteroscedastic within-group errors Estimate a separate variance for each level of a factor. We specify that the mixed model should estimate a separate variance for each level of density.
turnip.lme.het = lme(sqrt(yield)~gen*density, random=~1|block, data=turnip, weights=varIdent(form=~1|density))
turnip.lme.het
## Linear mixed-effects model fit by REML
## Data: turnip
## Log-restricted-likelihood: -67.33769
## Fixed: sqrt(yield) ~ gen * density
## (Intercept) genMarco density2 density4
## 1.61393154 -0.48964237 0.40986169 1.12970894
## density8 genMarco:density2 genMarco:density4 genMarco:density8
## 1.36488918 -0.06774089 -0.04907923 0.15760443
##
## Random effects:
## Formula: ~1 | block
## (Intercept) Residual
## StdDev: 0.2332858 0.4805598
##
## Variance function:
## Structure: Different standard deviations per stratum
## Formula: ~1 | density
## Parameter estimates:
## 1 2 4 8
## 1.0000000 0.9609899 2.1941207 1.8086482
## Number of Observations: 64
## Number of Groups: 4
plot(turnip.lme.het)
The unequal variance problem seems to be completely solved. We can now use this model to proceed with the analysis.
anova(turnip.lme.het, type='marginal')
Conclude that the interaction effect is not needed and simplify the model:
turnip.lme.het.add = lme(sqrt(yield)~gen+density, random=~1|block, data=turnip, weights=varIdent(form=~1|density))
anova(turnip.lme.het.add)
Both main effects are sigificant we could proceed with further analysis!
If we have designs with plot structure, we have to tell the aov() function about the plot structure otherwise the wrong F tests are produced.
summary(aov(sqrt(count)~spray+Error(block), data=InsectSprays))
##
## Error: block
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 5 5.554 1.111
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## spray 5 88.44 17.688 52.62 <2e-16 ***
## Residuals 61 20.50 0.336
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The Error term is used to tell the aov() to add a random block effect, as a result no test for the significance of the block effect is performed, as this is usually not of interest. Or we could fit with lme and get the same results.
library(nlme)
ins.lme=lme(sqrt(count)~spray, random=~1|block, data=InsectSprays)
anova(ins.lme, type='marginal')
Use aov() just for balanced designs and lme() for a more general approach.
If block sizes are smaller than the number of treatments, complete block designs are not realizable. Instead we use a binary design, this means in each block one treatment is either observed once or not at all. And all treatments are observed the same number of times.
Definition: Each of the v treatments is observed r times , and each of the b blocks has k plots (uniformity). Also the design is binary and each pair of treatments appears together \(\lambda\) times (number of concurrences).
Conditions (necessary not sufficient): 1. The amount of treatments times the amount of times each treatment is observed is equal to the amount of blocks times the amount of plots: v\(\times\)r = b\(\times\)k 2. r(k-1) = \(\lambda\)\(\times\)(v-1), each treatment is applied in r blocks and in each of these blocks are k-1 other treatments. Since each of the treatments is exactly observed \(\lambda\) times together with of the v-1 other treatments, the condition follows. 3. b \(\le\) v
library(ibd)
set.seed(1)
bibd(7, 7, 3, 3, 1)$design
## [,1] [,2] [,3]
## Block-1 1 4 6
## Block-2 5 6 7
## Block-3 3 4 5
## Block-4 1 3 7
## Block-5 2 4 7
## Block-6 2 3 6
## Block-7 1 2 5
We have bibd(v, b, r, k, lambda, ntrail, pbar=FALSE)
detergent = read.table('detergent.csv', header=TRUE, sep=',')
head(detergent)
addmargins(with(detergent, table(treatment, block))) #shows design
## block
## treatment B01 B02 B03 B04 B05 B06 B07 B08 B09 B10 B11 B12 Sum
## T1 0 0 0 1 0 0 0 1 1 0 0 1 4
## T2 0 1 0 0 1 0 0 0 0 1 0 1 4
## T3 1 0 1 0 0 0 0 0 0 0 1 1 4
## T4 1 1 0 0 0 1 0 1 0 0 0 0 4
## T5 0 0 0 1 0 1 0 0 0 1 1 0 4
## T6 0 0 1 0 1 1 0 0 1 0 0 0 4
## T7 0 0 0 0 1 0 1 1 0 0 1 0 4
## T8 1 0 0 0 0 0 1 0 1 1 0 0 4
## T9 0 1 1 1 0 0 1 0 0 0 0 0 4
## Sum 3 3 3 3 3 3 3 3 3 3 3 3 36
ggplot(detergent, aes(treatment, plates)) + geom_jitter(shape=4, width=0.1)
We have a fixed treatment effect, a fixed block effect and a random error term. The model assumes no interaction effects between block and treatments. Question: Is there a treatment effect after adjusting for any block effect? (The order of the variables is important, block comes first for this analysis!)
detergent.lm = lm(plates~block+treatment, data=detergent)
anova(detergent.lm)
This strategy is sometimes called intrablock analysis because we base the analysis on differences from within the block. The treatment effect adjusted for blocks is highly significant, now we could proceed with pairwise comparisons or treatment contrasts.
Adjusting for block effects Adjust the observation by the difference of the least square estimate of the effect of the block and the least squares estimate of the mean block effect.
theta.hat = c(0, coef(detergent.lm)[2:12])
theta.avg = mean(theta.hat)
detergent$block.num = as.factor(detergent$block)
detergent$adj = theta.hat[detergent$block.num] - theta.avg
detergent$plates.adj = detergent$plates - detergent$adj
head(detergent)
ggplot(detergent, aes(treatment, plates.adj)) + geom_jitter(shape=4, width=0.1)
The adjusted values do not differ a lot from the original values, since the treatment effects dominate block effect for the data set.
Block effects are modeled as random effects.
detergent.lme = lme(plates~treatment, random=~1|block, data=detergent)
anova(detergent.lme)
#split for each block, colors show variety
ggplot(Oats, aes(x=nitro, y=yield, col=Variety)) + geom_point() + geom_line() + facet_wrap(~Block)
#split for each variety, grouped according to block
ggplot(Oats, aes(x=nitro, y=yield, group=Block)) + geom_point() + geom_line() + facet_wrap(~Variety)
Classically, split plot designs are analyzed with nested random effects. The default treatment reference levels are the first levels of both factors. We have the two fixed main effects of the two factors and their interaction effect. A random intercept for each block, as well as a random intercept for each plot within the block and a random error term.
We assume that the random effects and the error terms are independent.
library(nlme)
Oats$nitroF = factor(Oats$nitro) #turn the nitrogen levels into factors
Oats.lme = lme(yield~Variety*nitroF, random=~1|Block/Variety, data=Oats)
summary(Oats.lme)
## Linear mixed-effects model fit by REML
## Data: Oats
## AIC BIC logLik
## 559.0285 590.4437 -264.5143
##
## Random effects:
## Formula: ~1 | Block
## (Intercept)
## StdDev: 14.64496
##
## Formula: ~1 | Variety %in% Block
## (Intercept) Residual
## StdDev: 10.29863 13.30727
##
## Fixed effects: yield ~ Variety * nitroF
## Value Std.Error DF t-value p-value
## (Intercept) 80.00000 9.106958 45 8.784492 0.0000
## VarietyMarvellous 6.66667 9.715028 10 0.686222 0.5082
## VarietyVictory -8.50000 9.715028 10 -0.874933 0.4021
## nitroF0.2 18.50000 7.682957 45 2.407927 0.0202
## nitroF0.4 34.66667 7.682957 45 4.512152 0.0000
## nitroF0.6 44.83333 7.682957 45 5.835427 0.0000
## VarietyMarvellous:nitroF0.2 3.33333 10.865342 45 0.306786 0.7604
## VarietyVictory:nitroF0.2 -0.33333 10.865342 45 -0.030679 0.9757
## VarietyMarvellous:nitroF0.4 -4.16667 10.865342 45 -0.383482 0.7032
## VarietyVictory:nitroF0.4 4.66667 10.865342 45 0.429500 0.6696
## VarietyMarvellous:nitroF0.6 -4.66667 10.865342 45 -0.429500 0.6696
## VarietyVictory:nitroF0.6 2.16667 10.865342 45 0.199411 0.8428
## Correlation:
## (Intr) VrtyMr VrtyVc ntF0.2 ntF0.4 ntF0.6 VM:F0.2
## VarietyMarvellous -0.533
## VarietyVictory -0.533 0.500
## nitroF0.2 -0.422 0.395 0.395
## nitroF0.4 -0.422 0.395 0.395 0.500
## nitroF0.6 -0.422 0.395 0.395 0.500 0.500
## VarietyMarvellous:nitroF0.2 0.298 -0.559 -0.280 -0.707 -0.354 -0.354
## VarietyVictory:nitroF0.2 0.298 -0.280 -0.559 -0.707 -0.354 -0.354 0.500
## VarietyMarvellous:nitroF0.4 0.298 -0.559 -0.280 -0.354 -0.707 -0.354 0.500
## VarietyVictory:nitroF0.4 0.298 -0.280 -0.559 -0.354 -0.707 -0.354 0.250
## VarietyMarvellous:nitroF0.6 0.298 -0.559 -0.280 -0.354 -0.354 -0.707 0.500
## VarietyVictory:nitroF0.6 0.298 -0.280 -0.559 -0.354 -0.354 -0.707 0.250
## VV:F0.2 VM:F0.4 VV:F0.4 VM:F0.6
## VarietyMarvellous
## VarietyVictory
## nitroF0.2
## nitroF0.4
## nitroF0.6
## VarietyMarvellous:nitroF0.2
## VarietyVictory:nitroF0.2
## VarietyMarvellous:nitroF0.4 0.250
## VarietyVictory:nitroF0.4 0.500 0.500
## VarietyMarvellous:nitroF0.6 0.250 0.500 0.250
## VarietyVictory:nitroF0.6 0.500 0.250 0.500 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.81300898 -0.56144838 0.01758044 0.63864476 1.57034166
##
## Number of Observations: 72
## Number of Groups:
## Block Variety %in% Block
## 6 18
The standard deviation of the block is given with 14.645, for the variety inside the blocks with 10.298 and for the error terms with 13.307. To test the significance of the fixed effects we are using the marginal F test approach.
anova(Oats.lme, type='marginal')
The interaction effect is not significant! No variety specific effect of the nitrogen level can be detected. We can remove the interaction effect and fit the model again.
Fit the model again without the non significant interaction effect.
Oats.lme = lme(yield~Variety+nitroF, random=~1|Block/Variety, data=Oats)
summary(Oats.lme)
## Linear mixed-effects model fit by REML
## Data: Oats
## AIC BIC logLik
## 586.0688 605.7756 -284.0344
##
## Random effects:
## Formula: ~1 | Block
## (Intercept)
## StdDev: 14.64488
##
## Formula: ~1 | Variety %in% Block
## (Intercept) Residual
## StdDev: 10.47345 12.74986
##
## Fixed effects: yield ~ Variety + nitroF
## Value Std.Error DF t-value p-value
## (Intercept) 79.91667 8.220351 51 9.721807 0.0000
## VarietyMarvellous 5.29167 7.078910 10 0.747526 0.4720
## VarietyVictory -6.87500 7.078910 10 -0.971195 0.3544
## nitroF0.2 19.50000 4.249955 51 4.588284 0.0000
## nitroF0.4 34.83333 4.249955 51 8.196166 0.0000
## nitroF0.6 44.00000 4.249955 51 10.353051 0.0000
## Correlation:
## (Intr) VrtyMr VrtyVc ntF0.2 ntF0.4
## VarietyMarvellous -0.431
## VarietyVictory -0.431 0.500
## nitroF0.2 -0.259 0.000 0.000
## nitroF0.4 -0.259 0.000 0.000 0.500
## nitroF0.6 -0.259 0.000 0.000 0.500 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.84134146 -0.66279747 -0.06694274 0.63822469 1.66066801
##
## Number of Observations: 72
## Number of Groups:
## Block Variety %in% Block
## 6 18
anova(Oats.lme, type='marginal')
The F test shows that the main effect of the Variety is not significant. This does not mean that we completely remove the Variety from the model, we remove the main effect but still keep it as a random effect, since it models intra-strip correlation within each block. The fixed effect models a systematic variety difference across blocks since it is not specific to a particular block.
Fit the model again without the main effect for Variety:
Oats.lme = lme(yield~nitroF, random=~1|Block/Variety, data=Oats)
summary(Oats.lme)
## Linear mixed-effects model fit by REML
## Data: Oats
## AIC BIC logLik
## 596.2187 611.7553 -291.1094
##
## Random effects:
## Formula: ~1 | Block
## (Intercept)
## StdDev: 14.50594
##
## Formula: ~1 | Variety %in% Block
## (Intercept) Residual
## StdDev: 11.03866 12.74987
##
## Fixed effects: yield ~ nitroF
## Value Std.Error DF t-value p-value
## (Intercept) 79.38889 7.132392 51 11.130753 0
## nitroF0.2 19.50000 4.249957 51 4.588282 0
## nitroF0.4 34.83333 4.249957 51 8.196161 0
## nitroF0.6 44.00000 4.249957 51 10.353046 0
## Correlation:
## (Intr) ntF0.2 ntF0.4
## nitroF0.2 -0.298
## nitroF0.4 -0.298 0.500
## nitroF0.6 -0.298 0.500 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.78155507 -0.61168963 0.02222435 0.62200776 1.68137700
##
## Number of Observations: 72
## Number of Groups:
## Block Variety %in% Block
## 6 18
anova(Oats.lme)
So far we ignored the ordered structure of the nitrogen factor. Since all levels are equidistant, we can treat the levels as numeric. To take advantage of the ordered structure we could convert the nitrogen into an ordered factor. Here we fit the model again with no fixed effect for Variety:
Oats$nitroF = factor(Oats$nitro, ordered=TRUE)
Oats.lme.o = lme(yield~nitroF, random=~1|Block/Variety, data=Oats)
coef(summary(Oats.lme.o))
## Value Std.Error DF t-value p-value
## (Intercept) 103.9722222 6.640669 51 15.6568891 4.030058e-21
## nitroF.L 32.9447349 3.005169 51 10.9626912 5.121203e-15
## nitroF.Q -5.1666667 3.005169 51 -1.7192602 9.163171e-02
## nitroF.C -0.4472136 3.005169 51 -0.1488148 8.822866e-01
If we use an ordered structure so called orthogonal polynomial contrasts are used for the nitrogen factor. The first contrast estimates the linear trend, the second the quadratic effect orthogonal (independent) of the linear trend, and the third estimates the orthogonal cubic trend. We can see that only the linear trend is significant. We want to look again at the fitted values (excluding random effects) from the model that treats nitrogen as a factor without ordered structure:
df = data.frame(nitroF=levels(Oats$nitroF))
df$pred = predict(Oats.lme, df, level=0)
df
Here the linearity seem to be reflected except maybe for the last level. We can now fit our last model with a linear regression structure for the fixed effect of nitrogen:
Oats.lme.lin = lme(yield~nitro, random=~1|Block/Variety, data=Oats)
summary(Oats.lme.lin)
## Linear mixed-effects model fit by REML
## Data: Oats
## AIC BIC logLik
## 603.0418 614.2842 -296.5209
##
## Random effects:
## Formula: ~1 | Block
## (Intercept)
## StdDev: 14.50598
##
## Formula: ~1 | Variety %in% Block
## (Intercept) Residual
## StdDev: 11.00467 12.86696
##
## Fixed effects: yield ~ nitro
## Value Std.Error DF t-value p-value
## (Intercept) 81.87222 6.945280 53 11.78818 0
## nitro 73.66667 6.781483 53 10.86291 0
## Correlation:
## (Intr)
## nitro -0.293
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.74380777 -0.66475224 0.01710434 0.54298790 1.80298914
##
## Number of Observations: 72
## Number of Groups:
## Block Variety %in% Block
## 6 18
We can now interpret that an increase of the nitrogen level of 0.2 units rises the yield by 0.2$$73.66 = 14.73 units on average for a typical plot. Visualization:
Oats$yield.factor = predict(Oats.lme)
Oats$yield.lin = predict(Oats.lme.lin)
ggplot(Oats, aes(x = nitro, y = yield, col = Variety)) + geom_point() + geom_line(linetype = "solid") + geom_line(aes(y = yield.factor, col = Variety), linetype = "dotted") + geom_line(aes(y = yield.lin, col = Variety), linetype = "dashed") + facet_wrap(~Block)
The factor fitted model is represented by the solid lines, the linear regression model fitted values by dashed lines. The two models seem to make very similar predictions for the fitted values, if that is the case the linear model is to be preferred since less parameters have to be estimated (bias-variance trade off).
We have five blocks depending on the distance to a hedge, in each block one plot is close to the forest and one further away (the variable location acts as a treatment here). One person should do the measurements for one block, to cancel out the observer bias. Strictly speaking this is not an experiment, because of the missing randomization, but an observational study. We want to examine the location effect (treatment) on the shannon index.
gasel = read.table('gasel.csv', sep=',', header=TRUE)
head(gasel)
ggplot(gasel, aes(block, shannon, col=location)) + geom_point()
ggplot(gasel, aes(location, shannon, col=location)) + geom_point() + facet_grid(~block)
The Shannon index (which measures biodiversity) is higher in open plots. Is the effect significant? Since we do not have many samples a non parametric test seems to be the right choice for analysation of the data.
friedman.test(shannon~location|block, data=gasel)
##
## Friedman rank sum test
##
## data: shannon and location and block
## Friedman chi-squared = 5, df = 1, p-value = 0.02535
The Friedmans test finds a significant treatment, here location, effect. To use covariates we use a parametric mixed effects model (although the data sample is perhaps too small).
gasel.blk = lme(shannon~location, random=~1|block, data=gasel)
coef(summary(gasel.blk))
## Value Std.Error DF t-value p-value
## (Intercept) 2.768 0.11968713 4 23.126965 0.0000207149
## locationshaded -0.444 0.07978719 4 -5.564803 0.0051073584
The mixed effects model also shows a significant location effect (open plots are the reference level) and we get an estimate for the effect. For a typical block the Shannon index is of the open plots is 2.77 and the Shannon is on average 0.44 units lower in the shaded plots than in the open plots.
We now want to also take into account the effect of the radiation (Wh/m2) data on the Shannon index. Visualization:
gasel$radiation = gasel$radiation/100000
ggplot(gasel, aes(radiation, shannon)) + geom_point(aes(col=location)) + geom_line(aes(group=block))
The two plots of each block are connected by line. In each block a clear effect of radiation is seen. We now fit a model and allow two different lienar regression lines to be fit for shaded and open plots:
gasel.lme = lme(shannon~radiation*location, random=~1|block, data=gasel)
coef(summary(gasel.lme))
## Value Std.Error DF t-value p-value
## (Intercept) 1.95111780 1.5909478 4 1.2263871 0.2873091
## radiation 0.20860938 0.4053724 2 0.5146117 0.6580502
## locationshaded -0.26602504 1.4582940 2 -0.1824221 0.8720680
## radiation:locationshaded 0.04623994 0.3653919 2 0.1265489 0.9108725
ggplot(gasel, aes(radiation, shannon, col=location)) + geom_point() + geom_smooth(method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'
The slope for the shaded plots is 0.046 units higher than the slope of the open locations, but the slope difference is not significant (interaction effect). So we can fit a model without it, but still allow for different intercepts:
gasel.lme = lme(shannon~radiation+location, random=~1 | block, data=gasel)
coef(summary(gasel.lme))
## Value Std.Error DF t-value p-value
## (Intercept) 1.8180334 0.6181093 4 2.9412813 0.04233508
## radiation 0.2425955 0.1557267 3 1.5578284 0.21715256
## locationshaded -0.1022203 0.2366051 3 -0.4320293 0.69488937
ggplot(gasel, aes(radiation, shannon, col=location)) + geom_point() + geom_smooth(aes(y = predict(gasel.lme, gasel)), method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'
The shaded locations have an intercept which is 0.1 units lower than the one for the open locations, but again the difference is not significant. Further reduction to just one common regression line:
gasel.lme = lme(shannon~radiation, random=~1|block, data=gasel)
coef(summary(gasel.lme))
## Value Std.Error DF t-value p-value
## (Intercept) 1.5625107 0.20123014 4 7.764794 0.001482769
## radiation 0.3062472 0.05719127 4 5.354789 0.005866770
fixef(gasel.lme)
## (Intercept) radiation
## 1.5625107 0.3062472
ggplot(gasel, aes(radiation, shannon)) + geom_point() + geom_smooth(aes(y=predict(gasel.lme, gasel)), method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'
For this data set the covariate was so important that it completely eliminated the treatment from the model.
data("PlantGrowth")
head(PlantGrowth)
with(PlantGrowth, tapply(weight, group, length))
## ctrl trt1 trt2
## 10 10 10
with(PlantGrowth, tapply(weight, group, summary))
## $ctrl
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.170 4.550 5.155 5.032 5.293 6.110
##
## $trt1
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3.590 4.207 4.550 4.661 4.870 6.030
##
## $trt2
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.920 5.268 5.435 5.526 5.735 6.310
with(PlantGrowth, tapply(weight, group, sd))
## ctrl trt1 trt2
## 0.5830914 0.7936757 0.4425733
Each of the three treatments has the same amount of observations, 10 per treatment. The range of the values is the biggest for treatment 1, which has the smallest mean with 4.66 and the highest standard deviation with 0.79, the control treatment has a mean of 5.03 and a standard deviation of 0.58, the treatment 2 has a mean of 5.52 and a standard deviation of 0.44.
library(ggplot2)
ggplot(PlantGrowth, aes(group, weight)) + geom_jitter(width=0.1, height=0)
We obtain a strip plot of the data. We can see that the group of trt1 has two data points which are quite far away from the other data points, this also explains the high standard deviation.
plant.lm = lm(weight~group, data=PlantGrowth)
anova(plant.lm)
The parametric overall F test indicates a significant treatment effect, i.e. the null hypothesis of equal group means can be rejected for a significance level of \(\alpha\) = 0.05, with a p-value of 0.0159.
plant.aov = aov(weight~group, data=PlantGrowth)
TukeyHSD(plant.aov)
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = weight ~ group, data = PlantGrowth)
##
## $group
## diff lwr upr p adj
## trt1-ctrl -0.371 -1.0622161 0.3202161 0.3908711
## trt2-ctrl 0.494 -0.1972161 1.1852161 0.1979960
## trt2-trt1 0.865 0.1737839 1.5562161 0.0120064
We do not have a significant difference for the control treatment mean to either of the two treatment means, but we do have a significant difference between the two treatment means (p-value = 0.012). With this method of pairwise comparisons the familywise error rate is conrtolled at 5%.
library(multcomp)
summary(glht(plant.lm, mcp(group='Tukey')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Tukey Contrasts
##
##
## Fit: lm(formula = weight ~ group, data = PlantGrowth)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## trt1 - ctrl == 0 -0.3710 0.2788 -1.331 0.391
## trt2 - ctrl == 0 0.4940 0.2788 1.772 0.198
## trt2 - trt1 == 0 0.8650 0.2788 3.103 0.012 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
The HSD test of the multcomp package also just finds a significant difference in means between treatment 1 and treatment 2. We can also say that trt2 produces a weight which is on average 0.865 units higher than trt1.
summary(glht(plant.lm, mcp(group='Dunnet')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Dunnett Contrasts
##
##
## Fit: lm(formula = weight ~ group, data = PlantGrowth)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## trt1 - ctrl == 0 -0.3710 0.2788 -1.331 0.323
## trt2 - ctrl == 0 0.4940 0.2788 1.772 0.153
## (Adjusted p values reported -- single-step method)
As before we do not have a significant difference for the treatment means compared to the control treatment. We can see that the p-values are a bit smaller than for Tukey’s HSD since the multiplicity adjustment is more strict for three test than for two tests.
library(car)
qqPlot(resid(plant.lm))
## [1] 17 15
shapiro.test(resid(plant.lm))
##
## Shapiro-Wilk normality test
##
## data: resid(plant.lm)
## W = 0.96607, p-value = 0.4379
Visually we do have a slightly heavy upper tail, but this could also just be due to chance. The null hypothesis of normality is not rejected by the shapiro test, so we do not detected any problems with non normality.
plot(fitted(plant.lm), resid(plant.lm), las=1, xlab='Fitted values', ylab='Residuals')
abline(h=0)
#or
plot(plant.lm, which=1)
The variance of the residuals seems to decrease with an increase of the fitted values. To check the variance further we could perform a test.
bartlett.test(weight~group, data=PlantGrowth)
##
## Bartlett test of homogeneity of variances
##
## data: weight by group
## Bartlett's K-squared = 2.8786, df = 2, p-value = 0.2371
The Bartlett does not reject the null hypothesis of equal variance (p-value = 0.2371), so we conclude that there a no problems with heteroscedasticity.
kruskal.test(weight~group, data=PlantGrowth)
##
## Kruskal-Wallis rank sum test
##
## data: weight by group
## Kruskal-Wallis chi-squared = 7.9882, df = 2, p-value = 0.01842
The Kruskal Wallis test is significant, that means that the null hypothesis of equal underlying weight distributions is rejected and at least two treatment effects differ. In conclusion a significant effect (p-value = 0.01842) of the treatment on teh weight distribution is found.
library(asbio)
with(PlantGrowth, BDM.test(weight, group))
##
## One way Brunner-Dette-Munk test
##
## df1 df2 F* P(F > F*)
## 1.873924 23.7978 5.132413 0.01546714
#or
with(PlantGrowth, trim.test(weight, group))
## $Results
## df1 df2 F* P(>F)
## 1 2 9.639328 8.372075 0.007806059
The Brunner-Dette-Munkrejects the null hypothesis that the distribution of the weights are the same in each group (p-value = 0.0155).
The Trim test finds a significant difference of the 20% trimmed means in the three groups (p-value = 0.0078).
library(lmPerm)
set.seed(42)
anova(lmp(weight~group, data=PlantGrowth))
## [1] "Settings: unique SS "
## Analysis of Variance Table
##
## Response: weight
## Df R Sum Sq R Mean Sq Iter Pr(Prob)
## group 2 3.7663 1.8832 5000 0.009 **
## Residuals 27 10.4921 0.3886
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The Permutation test has a significant Monte Carlo p-value with 0.009 and we can conclude that the null hypothesis of equal treatment means is rejected.
data('morley')
head(morley)
with(morley, tapply(Speed, Expt, length))
## 1 2 3 4 5
## 20 20 20 20 20
morley$Expt = as.factor(morley$Expt)
morley.lm = lm(Speed~Expt, data=morley)
library(car)
Anova(morley.lm, type=2)
The overall F test is significant (p-value = 0.003114), hence we can reject the null hypothesis of equal means for each experiment.
library(ggplot2)
ggplot(morley, aes(Expt, Speed)) + geom_jitter(width=0.1, height=0)
#or
plot(Speed~Expt, data=morley)
We can see that Experiment 1 has a high variance, and Experiment 3 has three outliers, which could be problematic for the analysis.
library(agricolae)
HSD.test(morley.lm, 'Expt', group=TRUE, console=TRUE)
##
## Study: morley.lm ~ "Expt"
##
## HSD Test for Speed
##
## Mean Square Error: 5510.632
##
## Expt, means
##
## Speed std r Min Max
## 1 909.0 104.92604 20 650 1070
## 2 856.0 61.16414 20 760 960
## 3 845.0 79.10686 20 620 970
## 4 820.5 60.04165 20 720 920
## 5 831.5 54.21934 20 740 950
##
## Alpha: 0.05 ; DF Error: 95
## Critical Value of Studentized Range: 3.932736
##
## Minimun Significant Difference: 65.28006
##
## Treatments with the same letter are not significantly different.
##
## Speed groups
## 1 909.0 a
## 2 856.0 ab
## 3 845.0 ab
## 5 831.5 b
## 4 820.5 b
#or emmeans
library(emmeans)
ref_grid(morley.lm)
## 'emmGrid' object with variables:
## Expt = 1, 2, 3, 4, 5
morley.emm = emmeans(morley.lm, 'Expt')
morley.emm
## Expt emmean SE df lower.CL upper.CL
## 1 909 16.6 95 876 942
## 2 856 16.6 95 823 889
## 3 845 16.6 95 812 878
## 4 820 16.6 95 788 853
## 5 832 16.6 95 799 864
##
## Confidence level used: 0.95
pairs(morley.emm)
## contrast estimate SE df t.ratio p.value
## 1 - 2 53.0 23.5 95 2.258 0.1680
## 1 - 3 64.0 23.5 95 2.726 0.0575
## 1 - 4 88.5 23.5 95 3.770 0.0026
## 1 - 5 77.5 23.5 95 3.301 0.0116
## 2 - 3 11.0 23.5 95 0.469 0.9900
## 2 - 4 35.5 23.5 95 1.512 0.5572
## 2 - 5 24.5 23.5 95 1.044 0.8343
## 3 - 4 24.5 23.5 95 1.044 0.8343
## 3 - 5 13.5 23.5 95 0.575 0.9784
## 4 - 5 -11.0 23.5 95 -0.469 0.9900
##
## P value adjustment: tukey method for comparing a family of 5 estimates
The output indicates a significant difference in mean light speed between Experiment 1 and 4 and Experiment 1 and 5.
library(car)
qqPlot(resid(morley.lm))
## 014 047
## 14 47
We can see that the distribution has a heavy lower tail. The qqPlot shows problems with nonnormality. For further assessment one could perform a Shapiro test:
shapiro.test(resid(morley.lm))
##
## Shapiro-Wilk normality test
##
## data: resid(morley.lm)
## W = 0.96779, p-value = 0.01501
The null hypothesis of normaility is rejected with a p-value of 0.01501.
bartlett.test(Speed~Expt, data=morley)
##
## Bartlett test of homogeneity of variances
##
## data: Speed by Expt
## Bartlett's K-squared = 11.552, df = 4, p-value = 0.02102
leveneTest(Speed~Expt, data=morley)
plot(morley.lm, which=1)
The Bartlett test has a p-value of 0.02102 and we would have to reject the null hypothesis of equal variance. The Levene’s Test does not reject the null hypothesis of equal variance, this could be due to the fact that the Bartlett test is very sensitive to non normality. The residual vs. fitted plot also shows some problems with variance, since it increases with the fitted values.
kruskal.test(Speed~Expt, data=morley)
##
## Kruskal-Wallis rank sum test
##
## data: Speed by Expt
## Kruskal-Wallis chi-squared = 15.022, df = 4, p-value = 0.004656
#or use a robust method
library(asbio)
with(morley, BDM.test(Speed, Expt))
##
## One way Brunner-Dette-Munk test
##
## df1 df2 F* P(F > F*)
## 3.819825 89.37885 4.248446 0.003906571
We perform a Kruskal-Wallis test even though the test suffers if we do not have equal variances. We reject the null hypothesis that the Experiments are coming from the same distribution (p-value = 0.004656). The Brunner Dette Munk test comes to the same conclusion (p-value = 0.0039).
The yan.winterwheat data frame from the agridat library contains data on the yield (in tons per ha) of 18 genotypes (variable gen, this means varieties) of winter wheat grown in 9 environments (variable env, these are locations). Each variety was randomly allocated to exactly one plot for each location. Our aim to model the effect of the different genotypes and the different environments on the yield.
library(agridat)
data('yan.winterwheat')
head(yan.winterwheat)
unique(yan.winterwheat$gen)
## [1] Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc m12 Reb Ron Rub Zav
## 18 Levels: Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron ... m12
with(yan.winterwheat, tapply(yield, gen, length))
## Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron Rub Zav m12
## 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
with(yan.winterwheat, table(env, gen))
## gen
## env Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron Rub Zav m12
## BH93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## EA93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## HW93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## ID93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## KE93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## NN93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## OA93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## RN93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## WP93 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
18*9
## [1] 162
We have 18 different treatments which grow once per each of the nine environments. The treatments are the 18$$9 = 162 combinations of genotype and environment. Each treatment is observed exactly one time.
Since we are not interested in the fixed effect of the environments, but only in the random effects we should fit a mixed effect model to the data. With genotype being the fixed effect and environment a random effect, since we only observe each treatment once we cannot model the interaction between genotype and environment.
library(nlme)
ww.lme = lme(yield~gen, random=~1|env, data=yan.winterwheat)
anova(ww.lme, type='marginal')
I use anova to perform an overall F test, the result is a significant genotype effect (p-value < 0.0001).
summary(ww.lme)
## Linear mixed-effects model fit by REML
## Data: yan.winterwheat
## AIC BIC logLik
## 248.5094 307.9057 -104.2547
##
## Random effects:
## Formula: ~1 | env
## (Intercept) Residual
## StdDev: 0.8872641 0.3830808
##
## Fixed effects: yield ~ gen
## Value Std.Error DF t-value p-value
## (Intercept) 3.999444 0.3221436 136 12.415098 0.0000
## genAri 0.304667 0.1805860 136 1.687100 0.0939
## genAug 0.239556 0.1805860 136 1.326545 0.1869
## genCas 0.543889 0.1805860 136 3.011799 0.0031
## genDel 0.403111 0.1805860 136 2.232239 0.0272
## genDia 0.342444 0.1805860 136 1.896295 0.0600
## genEna -0.297889 0.1805860 136 -1.649568 0.1013
## genFun 0.674222 0.1805860 136 3.733524 0.0003
## genHam 0.433111 0.1805860 136 2.398364 0.0178
## genHar 0.460444 0.1805860 136 2.549723 0.0119
## genKar 0.282667 0.1805860 136 1.565274 0.1198
## genKat -0.780222 0.1805860 136 -4.320501 0.0000
## genLuc -0.205667 0.1805860 136 -1.138885 0.2568
## genReb 0.302667 0.1805860 136 1.676025 0.0960
## genRon 0.390222 0.1805860 136 2.160866 0.0325
## genRub 0.236889 0.1805860 136 1.311779 0.1918
## genZav 0.486333 0.1805860 136 2.693084 0.0080
## genm12 -0.457778 0.1805860 136 -2.534957 0.0124
## Correlation:
## (Intr) genAri genAug genCas genDel genDia genEna genFun genHam genHar
## genAri -0.28
## genAug -0.28 0.50
## genCas -0.28 0.50 0.50
## genDel -0.28 0.50 0.50 0.50
## genDia -0.28 0.50 0.50 0.50 0.50
## genEna -0.28 0.50 0.50 0.50 0.50 0.50
## genFun -0.28 0.50 0.50 0.50 0.50 0.50 0.50
## genHam -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genHar -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genKar -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genKat -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genLuc -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genReb -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genRon -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genRub -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genZav -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genm12 -0.28 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
## genKar genKat genLuc genReb genRon genRub genZav
## genAri
## genAug
## genCas
## genDel
## genDia
## genEna
## genFun
## genHam
## genHar
## genKar
## genKat 0.50
## genLuc 0.50 0.50
## genReb 0.50 0.50 0.50
## genRon 0.50 0.50 0.50 0.50
## genRub 0.50 0.50 0.50 0.50 0.50
## genZav 0.50 0.50 0.50 0.50 0.50 0.50
## genm12 0.50 0.50 0.50 0.50 0.50 0.50 0.50
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -3.2610953 -0.6274208 -0.0567675 0.5017550 2.5838475
##
## Number of Observations: 162
## Number of Groups: 9
#or
fixef(ww.lme)['(Intercept)'] + fixef(ww.lme)['genKat']
## (Intercept)
## 3.219222
The average yield of the reference level is 3.999444 and the yield of the Kat genotype is on average 0.780222 lower. We get an avergae yield for the Kat genotype of 3.219222 tons per ha.
# normality
library(car)
qqPlot(resid(ww.lme))
## OA93 RN93
## 121 139
shapiro.test(resid(ww.lme))
##
## Shapiro-Wilk normality test
##
## data: resid(ww.lme)
## W = 0.99089, p-value = 0.3886
plot(ww.lme)
We test for normality and equal variance of the error terms. Testing for normality of the random effect does not make sense since we only have 9 environments. Both the qqPlot and the shapiro test (p-value = 0.3886) do not show any problems with normality. The equal variance assumption for the error terms seems to be violated, the variance of the residuals seems to increase with the fitted values. A solution for this would be to model heteroscedastic within group errors, with a separate variance for each level of the gen factor.
ww.lme.het = lme(yield~gen, random=~1|env, data=yan.winterwheat, weights=varIdent(form=~1|gen))
plot(ww.lme.het)
friedman.test(yield~gen|env, yan.winterwheat)
##
## Friedman rank sum test
##
## data: yield and gen and env
## Friedman chi-squared = 75.222, df = 17, p-value = 2.671e-09
The Friedman test rejects the null hypothesis (p-value = 2.671e-09) that the treatment effect is the same for every level of the treatment. Loss of power if the data do come from a normal distribution!
The researcher correctly implements this idea as follows:
Kat.yield <- yan.winterwheat$yield[yan.winterwheat$gen == "Kat"]
as.numeric( t.test(Kat.yield, alternative = "two.sided")$conf.int)
## [1] 2.577343 3.861101
The true average yield of the Kat variety is between 2.577 and 3.861 tons per ha with a confidence of 95%
The approach ignores the random effect which is introduced by the different environments, hence evironment variability is not properly accounted for.
We use emmans:
library(emmeans)
ww.emm = emmeans(ww.lme, 'gen')
confint(ww.emm, adjust='none')
The true yield for a typical environment for the Kat genotyple lies between 2.48 and 3.96 tons/ha with a confidence of 95%.
“An experiment was conducted to compare two protective dyes for metal, both with each other and with ‘no dye’. Ten braided metal cords were broken into three pieces. The three pieces of each cord were randomly allocated to the three treatments. (. . . ) After the dyes had been applied, the cords were left to weather for a fixed time, then their strengths were measured.”
The file contains the strengths as a percentage of the nominal strength specification, it also contains the sums per cord and per treatment.
metal = read.csv('metal.csv', sep=',', dec='.', header=TRUE)
metal
The design is called a Randomized Complete Block design (RCBD) since every treatment (dyes) is allocated exactly one time in every plot and treatments are allocated randomly within blocks.
The data has to be converted to a tidy long format. First we exclude the row with the sum of all strength measurements per treatment, then we use the melt() function to convert the data frame to the long format. We choose the Cord as the id variable and melt the treatments which where stored as three separate columns into one column corresponding now to the treatment.
library(reshape2)
metal1 = metal[metal$Cord != 'Treatment total',]
mmetal = melt(metal1, id.vars = 'Cord', measure.vars = c("No.dye", "Dye.A", 'Dye.B'), variable.name='treatment', value.name='strength')
mmetal
We are interested in the fixed effects of the treatments, while accounting for the random effects of the blocks. We can fit a mixed effect model to the data. We assess the model assumptions of normality of the residuals and the equal variance assumption of the residuals. We do not assess the normality of the random intercepts since we only have 10 blocks.
metal.lme = lme(strength~treatment, random=~1|Cord, data=mmetal)
library(car)
qqPlot(resid(metal.lme))
## 5 9
## 15 29
shapiro.test(resid(metal.lme))
##
## Shapiro-Wilk normality test
##
## data: resid(metal.lme)
## W = 0.98163, p-value = 0.867
plot(metal.lme)
No problems with normality or equal variance can be detected.
summary(metal.lme)
## Linear mixed-effects model fit by REML
## Data: mmetal
## AIC BIC logLik
## 181.5369 188.0161 -85.76846
##
## Random effects:
## Formula: ~1 | Cord
## (Intercept) Residual
## StdDev: 4.333235 3.95704
##
## Fixed effects: strength ~ treatment
## Value Std.Error DF t-value p-value
## (Intercept) 96.67 1.855669 18 52.09441 0.0000
## treatmentDye.A 2.62 1.769642 18 1.48053 0.1560
## treatmentDye.B 4.95 1.769642 18 2.79718 0.0119
## Correlation:
## (Intr) trtD.A
## treatmentDye.A -0.477
## treatmentDye.B -0.477 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.85678766 -0.57511066 0.08508085 0.38943995 1.54081610
##
## Number of Observations: 30
## Number of Groups: 10
The standard deviation of the random intercept of the cord is given with 4.333235, while the standard deviation of the residuals is given with 3.95704. The importance of the blocking variable is reflected in the large standard deviation of the random intercept.
#first overall F test
anova(metal.lme, type='marginal')
#pairwise comparisons
library(emmeans)
metal.emm = emmeans(metal.lme, 'treatment')
metal.emm
## treatment emmean SE df lower.CL upper.CL
## No.dye 96.7 1.86 9 92.5 101
## Dye.A 99.3 1.86 9 95.1 103
## Dye.B 101.6 1.86 9 97.4 106
##
## Degrees-of-freedom method: containment
## Confidence level used: 0.95
pairs(metal.emm)
## contrast estimate SE df t.ratio p.value
## No.dye - Dye.A -2.62 1.77 18 -1.481 0.3233
## No.dye - Dye.B -4.95 1.77 18 -2.797 0.0305
## Dye.A - Dye.B -2.33 1.77 18 -1.317 0.4044
##
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 3 estimates
The overall F test finds a significant treatment effect. The average strength percentage of the control (no dye) is 96.7. The average strength percentage of Dye A is 99.3 and of Dye B 101.6. Only the difference of the control treatment and Dye B is significant (p-value = 0.0305). The p-value is adjusted to a familywise error rate of 5%.
paper = read.csv('paper.csv', sep=',', dec='.', header=TRUE)
head(paper)
str(paper)
## 'data.frame': 36 obs. of 4 variables:
## $ Replicate : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Pulp : int 1 1 1 1 2 2 2 2 3 3 ...
## $ Temperature: int 200 225 250 275 200 225 250 275 200 225 ...
## $ Strength : int 30 35 37 36 34 41 38 42 29 26 ...
paper$Replicate = factor(paper$Replicate)
paper$Pulp = factor(paper$Pulp)
paper$fTemperature = factor(paper$Temperature)
str(paper)
## 'data.frame': 36 obs. of 5 variables:
## $ Replicate : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
## $ Pulp : Factor w/ 3 levels "1","2","3": 1 1 1 1 2 2 2 2 3 3 ...
## $ Temperature : int 200 225 250 275 200 225 250 275 200 225 ...
## $ Strength : int 30 35 37 36 34 41 38 42 29 26 ...
## $ fTemperature: Factor w/ 4 levels "200","225","250",..: 1 2 3 4 1 2 3 4 1 2 ...
library(ggplot2)
ggplot(paper, aes(x=Temperature, y=Strength, col=Pulp)) + geom_point() + geom_line() + facet_wrap(~Replicate)
In the second Replicate for Temperature 275 and Pulp 1 there is an outlier, which is actually not a valid data point if we compare the point to the results in the paper.
paper$Strength[(paper$Replicate == '2') & (paper$Pulp) == '1' & (paper$Temperature) == '275'] = 41
paper$fTemperature = factor(paper$Temperature)
library(ggplot2)
ggplot(paper, aes(x=Temperature, y=Strength, col=Pulp)) + geom_point() + geom_line() + facet_wrap(~Replicate)
with(paper, tapply(Strength, list(Pulp, Temperature), mean))
## 200 225 250 275
## 1 29.66667 34.66667 39.33333 39.00000
## 2 33.33333 39.00000 39.66667 42.00000
## 3 30.66667 30.00000 34.66667 40.33333
with(paper, tapply(Strength, list(Pulp, Temperature), sd))
## 200 225 250 275
## 1 1.527525 2.516611 2.081666 2.645751
## 2 2.081666 2.645751 2.081666 2.000000
## 3 1.527525 4.000000 3.785939 4.509250
library(nlme)
paper.lme = lme(Strength ~ fTemperature*Pulp, random=~1|Replicate/Pulp, data=paper)
library(car)
qqPlot(resid(paper.lme))
## 2/2 2/1
## 19 16
plot(paper.lme)
No problems with the normality of the residuals or with the euqal vairance assumption.
summary(paper.lme)
## Linear mixed-effects model fit by REML
## Data: paper
## AIC BIC logLik
## 152.2556 169.9264 -61.1278
##
## Random effects:
## Formula: ~1 | Replicate
## (Intercept)
## StdDev: 1.573434
##
## Formula: ~1 | Pulp %in% Replicate
## (Intercept) Residual
## StdDev: 1.128851 1.993044
##
## Fixed effects: Strength ~ fTemperature * Pulp
## Value Std.Error DF t-value p-value
## (Intercept) 29.666667 1.604392 18 18.490908 0.0000
## fTemperature225 5.000000 1.627313 18 3.072549 0.0066
## fTemperature250 9.666667 1.627313 18 5.940262 0.0000
## fTemperature275 9.333333 1.627313 18 5.735425 0.0000
## Pulp2 3.666667 1.870210 4 1.960564 0.1215
## Pulp3 1.000000 1.870210 4 0.534699 0.6212
## fTemperature225:Pulp2 0.666667 2.301369 18 0.289683 0.7754
## fTemperature250:Pulp2 -3.333333 2.301369 18 -1.448414 0.1647
## fTemperature275:Pulp2 -0.666667 2.301369 18 -0.289683 0.7754
## fTemperature225:Pulp3 -5.666667 2.301369 18 -2.462303 0.0241
## fTemperature250:Pulp3 -5.666667 2.301369 18 -2.462303 0.0241
## fTemperature275:Pulp3 0.333333 2.301369 18 0.144841 0.8864
## Correlation:
## (Intr) fTm225 fTm250 fTm275 Pulp2 Pulp3 fT225:P2
## fTemperature225 -0.507
## fTemperature250 -0.507 0.500
## fTemperature275 -0.507 0.500 0.500
## Pulp2 -0.583 0.435 0.435 0.435
## Pulp3 -0.583 0.435 0.435 0.435 0.500
## fTemperature225:Pulp2 0.359 -0.707 -0.354 -0.354 -0.615 -0.308
## fTemperature250:Pulp2 0.359 -0.354 -0.707 -0.354 -0.615 -0.308 0.500
## fTemperature275:Pulp2 0.359 -0.354 -0.354 -0.707 -0.615 -0.308 0.500
## fTemperature225:Pulp3 0.359 -0.707 -0.354 -0.354 -0.308 -0.615 0.500
## fTemperature250:Pulp3 0.359 -0.354 -0.707 -0.354 -0.308 -0.615 0.250
## fTemperature275:Pulp3 0.359 -0.354 -0.354 -0.707 -0.308 -0.615 0.250
## fT250:P2 fT275:P2 fT225:P3 fT250:P3
## fTemperature225
## fTemperature250
## fTemperature275
## Pulp2
## Pulp3
## fTemperature225:Pulp2
## fTemperature250:Pulp2
## fTemperature275:Pulp2 0.500
## fTemperature225:Pulp3 0.250 0.250
## fTemperature250:Pulp3 0.500 0.250 0.500
## fTemperature275:Pulp3 0.250 0.500 0.500 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.1366311 -0.6873645 0.1481579 0.5103748 1.6541725
##
## Number of Observations: 36
## Number of Groups:
## Replicate Pulp %in% Replicate
## 3 9
anova(paper.lme, type='marginal')
The interaction effect of the temperature and the pulp is significant (p-value = 0.0271), hence we cannot simplify the model further.
We could exploit the ordered structure of the temperature factor:
paper$oTemperature = factor(paper$Temperature, ordered=TRUE)
paper.lme.o = lme(Strength~oTemperature*Pulp, random=~1|Replicate/Pulp, data=paper)
coef(summary(paper.lme.o))
## Value Std.Error DF t-value p-value
## (Intercept) 35.6666667 1.257385 18 28.36573760 2.154157e-16
## oTemperature.L 7.3044887 1.150684 18 6.34795219 5.570280e-06
## oTemperature.Q -2.6666667 1.150684 18 -2.31746165 3.245900e-02
## oTemperature.C -1.0434984 1.150684 18 -0.90685031 3.764665e-01
## Pulp2 2.8333333 1.229460 4 2.30453402 8.252652e-02
## Pulp3 -1.7500000 1.229460 4 -1.42338866 2.277160e-01
## oTemperature.L:Pulp2 -1.3416408 1.627313 18 -0.82445144 4.204745e-01
## oTemperature.Q:Pulp2 1.0000000 1.627313 18 0.61450982 5.465684e-01
## oTemperature.C:Pulp2 2.5342104 1.627313 18 1.55729716 1.368077e-01
## oTemperature.L:Pulp3 0.2236068 1.627313 18 0.13740857 8.922332e-01
## oTemperature.Q:Pulp3 5.8333333 1.627313 18 3.58464061 2.118345e-03
## oTemperature.C:Pulp3 0.0745356 1.627313 18 0.04580286 9.639717e-01
anova(paper.lme.o, type='marginal')
Only the cubic trend is not significant. The interaction effect of the pulp and treatment is still significant (p-value = 0.0271).
coef(summary(paper.lme))
## Value Std.Error DF t-value p-value
## (Intercept) 29.6666667 1.604392 18 18.4909081 3.714897e-13
## fTemperature225 5.0000000 1.627313 18 3.0725491 6.561433e-03
## fTemperature250 9.6666667 1.627313 18 5.9402616 1.273963e-05
## fTemperature275 9.3333333 1.627313 18 5.7354250 1.946204e-05
## Pulp2 3.6666667 1.870210 4 1.9605645 1.214760e-01
## Pulp3 1.0000000 1.870210 4 0.5346994 6.211966e-01
## fTemperature225:Pulp2 0.6666667 2.301369 18 0.2896827 7.753694e-01
## fTemperature250:Pulp2 -3.3333333 2.301369 18 -1.4484135 1.646977e-01
## fTemperature275:Pulp2 -0.6666667 2.301369 18 -0.2896827 7.753694e-01
## fTemperature225:Pulp3 -5.6666667 2.301369 18 -2.4623030 2.412202e-02
## fTemperature250:Pulp3 -5.6666667 2.301369 18 -2.4623030 2.412202e-02
## fTemperature275:Pulp3 0.3333333 2.301369 18 0.1448414 8.864456e-01
pred.df = expand.grid(fTemperature=levels(paper$fTemperature), Pulp=levels(paper$Pulp))
pred.df$fitted = predict(paper.lme, pred.df, level=0)
pred.df
ggplot(pred.df, aes(x=fTemperature, y=fitted, col=Pulp)) + geom_point() + geom_line(aes(group=Pulp))
library(emmeans)
paper.emm = emmeans(paper.lme, pairwise~Pulp|fTemperature)
pairs(paper.emm)
## $emmeans
## fTemperature = 200:
## contrast estimate SE df t.ratio p.value
## 1 - 2 -3.667 1.87 4 -1.961 0.2371
## 1 - 3 -1.000 1.87 4 -0.535 0.8593
## 2 - 3 2.667 1.87 4 1.426 0.4119
##
## fTemperature = 225:
## contrast estimate SE df t.ratio p.value
## 1 - 2 -4.333 1.87 4 -2.317 0.1639
## 1 - 3 4.667 1.87 4 2.495 0.1367
## 2 - 3 9.000 1.87 4 4.812 0.0186
##
## fTemperature = 250:
## contrast estimate SE df t.ratio p.value
## 1 - 2 -0.333 1.87 4 -0.178 0.9827
## 1 - 3 4.667 1.87 4 2.495 0.1367
## 2 - 3 5.000 1.87 4 2.673 0.1144
##
## fTemperature = 275:
## contrast estimate SE df t.ratio p.value
## 1 - 2 -3.000 1.87 4 -1.604 0.3436
## 1 - 3 -1.333 1.87 4 -0.713 0.7696
## 2 - 3 1.667 1.87 4 0.891 0.6735
##
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 3 estimates
##
## $contrasts
## fTemperature = 200:
## contrast estimate SE df t.ratio p.value
## (1 - 2) - (1 - 3) -2.667 1.87 4 -1.426 0.4119
## (1 - 2) - (2 - 3) -6.333 3.24 4 -1.955 0.2385
## (1 - 3) - (2 - 3) -3.667 1.87 4 -1.961 0.2371
##
## fTemperature = 225:
## contrast estimate SE df t.ratio p.value
## (1 - 2) - (1 - 3) -9.000 1.87 4 -4.812 0.0186
## (1 - 2) - (2 - 3) -13.333 3.24 4 -4.116 0.0316
## (1 - 3) - (2 - 3) -4.333 1.87 4 -2.317 0.1639
##
## fTemperature = 250:
## contrast estimate SE df t.ratio p.value
## (1 - 2) - (1 - 3) -5.000 1.87 4 -2.673 0.1144
## (1 - 2) - (2 - 3) -5.333 3.24 4 -1.646 0.3289
## (1 - 3) - (2 - 3) -0.333 1.87 4 -0.178 0.9827
##
## fTemperature = 275:
## contrast estimate SE df t.ratio p.value
## (1 - 2) - (1 - 3) -1.667 1.87 4 -0.891 0.6735
## (1 - 2) - (2 - 3) -4.667 3.24 4 -1.441 0.4058
## (1 - 3) - (2 - 3) -3.000 1.87 4 -1.604 0.3436
##
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 3 estimates
The predicted values show that the fitted values for the strength are the highest for pulp 2 for every temperature, we do not know if the differences are significant yet. We get the highest fitted value for pulp 2 and a temperature of 275, we use emmeans to test if the difference is significant. We do not get a significant difference for any of the pulps at 275 degrees, the only significant difference is between pulp 2 and three at 225 degrees (p-value = 0.0186).
mydata1 <- readRDS("tobacco_data.rds")
str(mydata1)
## 'data.frame': 56 obs. of 3 variables:
## $ block : Factor w/ 8 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 1 2 ...
## $ dose : Factor w/ 7 levels "0","250","500",..: 6 2 1 6 6 7 2 4 7 5 ...
## $ height: num 1299 1369 1170 1219 1120 ...
summary(mydata1)
## block dose height
## 1 : 7 0 :8 Min. : 607.6
## 2 : 7 250 :8 1st Qu.: 892.7
## 3 : 7 500 :8 Median : 999.7
## 4 : 7 1000:8 Mean :1012.5
## 5 : 7 1500:8 3rd Qu.:1126.7
## 6 : 7 2500:8 Max. :1421.1
## (Other):14 5000:8
table(mydata1$dose, mydata1$height)
##
## 607.6 627.6 667.8 671.9 776.4 787.1 797.8 827 844.2 846.5 852.4 853.6
## 0 0 0 0 0 0 0 0 0 0 0 0 0
## 250 0 0 0 0 0 1 0 0 0 1 0 0
## 500 0 0 0 0 0 0 0 0 0 0 0 1
## 1000 0 0 1 1 0 0 0 0 0 0 0 0
## 1500 0 1 0 0 0 0 0 0 1 0 0 0
## 2500 0 0 0 0 1 0 0 0 0 0 0 0
## 5000 1 0 0 0 0 0 1 1 0 0 1 0
##
## 873.4 875.9 898.3 917.9 947.4 947.6 960.4 960.7 968.7 971.7 972.2 975.5
## 0 0 0 1 1 0 0 0 0 0 1 0 0
## 250 0 0 0 0 0 0 0 0 0 0 0 0
## 500 1 0 0 0 0 0 1 0 0 0 1 0
## 1000 0 0 0 0 0 0 0 1 0 0 0 0
## 1500 0 0 0 0 1 0 0 0 0 0 0 0
## 2500 0 0 0 0 0 0 0 0 1 0 0 0
## 5000 0 1 0 0 0 1 0 0 0 0 0 1
##
## 975.8 990.2 993.8 999.4 1000 1003.3 1004 1006.2 1021.9 1031.5 1031.9
## 0 0 0 0 0 0 0 0 0 0 0 0
## 250 0 0 0 0 0 0 0 0 0 0 1
## 500 0 0 0 0 0 0 1 0 0 0 0
## 1000 0 0 0 0 1 0 0 1 0 0 0
## 1500 1 1 1 1 0 0 0 0 0 0 0
## 2500 0 0 0 0 0 1 0 0 1 0 0
## 5000 0 0 0 0 0 0 0 0 0 1 0
##
## 1069.7 1076.4 1083.7 1087.1 1093.3 1099.6 1120 1146.9 1169.5 1169.6
## 0 0 0 1 1 0 0 0 0 1 0
## 250 1 1 0 0 0 0 0 1 0 0
## 500 0 0 0 0 0 0 0 0 0 0
## 1000 0 0 0 0 0 1 0 0 0 1
## 1500 0 0 0 0 0 0 0 0 0 0
## 2500 0 0 0 0 1 0 1 0 0 0
## 5000 0 0 0 0 0 0 0 0 0 0
##
## 1172.6 1173.2 1174.9 1181.3 1219.1 1299.2 1322.4 1343.3 1369.2 1418.9
## 0 1 1 0 0 0 0 0 0 0 0
## 250 0 0 0 0 0 0 0 1 1 0
## 500 0 0 1 1 0 0 1 0 0 0
## 1000 0 0 0 0 0 0 0 0 0 1
## 1500 0 0 0 0 0 0 0 0 0 0
## 2500 0 0 0 0 1 1 0 0 0 0
## 5000 0 0 0 0 0 0 0 0 0 0
##
## 1421.1
## 0 0
## 250 0
## 500 0
## 1000 0
## 1500 1
## 2500 0
## 5000 0
with(mydata1, table(block, dose))
## dose
## block 0 250 500 1000 1500 2500 5000
## 1 1 1 1 1 1 1 1
## 2 1 1 1 1 1 1 1
## 3 1 1 1 1 1 1 1
## 4 1 1 1 1 1 1 1
## 5 1 1 1 1 1 1 1
## 6 1 1 1 1 1 1 1
## 7 1 1 1 1 1 1 1
## 8 1 1 1 1 1 1 1
with(mydata1, tapply(height, dose, mean))
## 0 250 500 1000 1500 2500 5000
## 1059.2500 1083.8750 1042.7750 999.3375 974.9375 1062.7375 864.4125
with(mydata1, tapply(height, dose, sd))
## 0 250 500 1000 1500 2500 5000
## 115.2085 207.0419 165.8663 248.9255 219.9814 160.6280 130.5180
library(ggplot2)
ggplot(mydata1, aes(dose, height)) + geom_point() + facet_grid(~block)
The spread of the data is very similar over the blocks, we cannot see a block effect.
mydata1.lme = lme(height~dose, random=~1 | block, data=mydata1)
anova(mydata1.lme, type='marginal')
We fit a mixed effect model to the data, with a fixed effect for the dose and a random effect for the block. The overall F test gives a non significant treatment effect (dose), the p-value is given with 0.199.
library(car)
qqPlot(resid(mydata1.lme))
## 1 2
## 49 50
plot(mydata1.lme)
Both the visual test for normality and the visual test for equal variance do not show any problems, we assume that those model assumptions hold.
cotton = readRDS('cotton_data.rds')
str(cotton)
## 'data.frame': 144 obs. of 6 variables:
## $ yield : num 0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
## $ year : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
## $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
## $ date : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ water : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
## $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...
c = cotton[cotton$year=='Y1', ]
str(c)
## 'data.frame': 72 obs. of 6 variables:
## $ yield : num 0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
## $ year : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
## $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
## $ date : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ water : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
## $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...
with(c, tapply(yield, list(date, nitrogen), mean))
## N0 N1
## D1 1.317778 2.773333
## D2 1.888889 3.074444
## D3 1.797778 2.664444
## D4 1.427778 1.735556
D2 has the highest average yield, with 3.074444 units.
cotton.lm = lm(yield~date*nitrogen, data=cotton)
Anova(cotton.lm, type=2)
The interaction effect of date and nitrogen is significant (p-value = 0.002), this means that the nitrogen effect depends on the sowing date.
cotton.lm3 = lm(yield~date*nitrogen*year, data=cotton)
Anova(cotton.lm3, type=2)
soy = read.delim('soybean.csv', header=TRUE, sep=';', dec='.')
str(soy)
## 'data.frame': 108 obs. of 4 variables:
## $ gen : chr "Tracy" "Centennial" "N72-137" "N72-3058" ...
## $ loc : chr "Plymouth" "Plymouth" "Plymouth" "Plymouth" ...
## $ block: chr "B1" "B1" "B1" "B1" ...
## $ yield: int 1307 1425 1289 1250 1546 1344 1280 1583 1656 1398 ...
soy$gen = factor(soy$gen)
soy$loc = factor(soy$loc)
soy$block = factor(soy$block)
str(soy)
## 'data.frame': 108 obs. of 4 variables:
## $ gen : Factor w/ 12 levels "Centennial","D74-7741",..: 12 1 3 4 5 10 2 7 8 9 ...
## $ loc : Factor w/ 3 levels "Clayton","Clinton",..: 3 3 3 3 3 3 3 3 3 3 ...
## $ block: Factor w/ 3 levels "B1","B2","B3": 1 1 1 1 1 1 1 1 1 1 ...
## $ yield: int 1307 1425 1289 1250 1546 1344 1280 1583 1656 1398 ...
with(soy, tapply(yield, gen, mean))
## Centennial D74-7741 N72-137 N72-3058 N72-3148 N73-1102 N73-693
## 1394.778 1406.444 1483.889 1330.667 1566.000 1501.444 1436.222
## N73-877 N73-882 R73-81 R75-12 Tracy
## 1403.333 1396.667 1373.778 1175.889 1369.889
The genotype N72-3148 produces the highest average yield with 1566.000.
soy.lm = lm(yield~gen*loc, data=soy)
Anova(soy.lm, type=2)
We use an overall F test, we find a significant effect of the genotype (p-value = 0.0004) and for the location (p-value < 0.00001). We do not find a significant interaction effect of genotype and location (p-value = 0.22).
This could be the case for a single observation but we can say that the average difference in yield is not significant.
soy.lm.n = lm(yield~gen+loc, data=soy)
library(emmeans)
soy.emm = emmeans(soy.lm.n, pairwise~gen)
pairs(soy.emm)
## $emmeans
## contrast estimate SE df t.ratio p.value
## Centennial - (D74-7741) -11.67 67.1 94 -0.174 1.0000
## Centennial - (N72-137) -89.11 67.1 94 -1.329 0.9732
## Centennial - (N72-3058) 64.11 67.1 94 0.956 0.9982
## Centennial - (N72-3148) -171.22 67.1 94 -2.553 0.3216
## Centennial - (N73-1102) -106.67 67.1 94 -1.590 0.9083
## Centennial - (N73-693) -41.44 67.1 94 -0.618 1.0000
## Centennial - (N73-877) -8.56 67.1 94 -0.128 1.0000
## Centennial - (N73-882) -1.89 67.1 94 -0.028 1.0000
## Centennial - (R73-81) 21.00 67.1 94 0.313 1.0000
## Centennial - (R75-12) 218.89 67.1 94 3.264 0.0638
## Centennial - Tracy 24.89 67.1 94 0.371 1.0000
## (D74-7741) - (N72-137) -77.44 67.1 94 -1.155 0.9910
## (D74-7741) - (N72-3058) 75.78 67.1 94 1.130 0.9925
## (D74-7741) - (N72-3148) -159.56 67.1 94 -2.379 0.4304
## (D74-7741) - (N73-1102) -95.00 67.1 94 -1.416 0.9574
## (D74-7741) - (N73-693) -29.78 67.1 94 -0.444 1.0000
## (D74-7741) - (N73-877) 3.11 67.1 94 0.046 1.0000
## (D74-7741) - (N73-882) 9.78 67.1 94 0.146 1.0000
## (D74-7741) - (R73-81) 32.67 67.1 94 0.487 1.0000
## (D74-7741) - (R75-12) 230.56 67.1 94 3.437 0.0392
## (D74-7741) - Tracy 36.56 67.1 94 0.545 1.0000
## (N72-137) - (N72-3058) 153.22 67.1 94 2.284 0.4947
## (N72-137) - (N72-3148) -82.11 67.1 94 -1.224 0.9857
## (N72-137) - (N73-1102) -17.56 67.1 94 -0.262 1.0000
## (N72-137) - (N73-693) 47.67 67.1 94 0.711 0.9999
## (N72-137) - (N73-877) 80.56 67.1 94 1.201 0.9877
## (N72-137) - (N73-882) 87.22 67.1 94 1.300 0.9771
## (N72-137) - (R73-81) 110.11 67.1 94 1.642 0.8888
## (N72-137) - (R75-12) 308.00 67.1 94 4.592 0.0008
## (N72-137) - Tracy 114.00 67.1 94 1.700 0.8639
## (N72-3058) - (N72-3148) -235.33 67.1 94 -3.509 0.0318
## (N72-3058) - (N73-1102) -170.78 67.1 94 -2.546 0.3254
## (N72-3058) - (N73-693) -105.56 67.1 94 -1.574 0.9141
## (N72-3058) - (N73-877) -72.67 67.1 94 -1.083 0.9947
## (N72-3058) - (N73-882) -66.00 67.1 94 -0.984 0.9977
## (N72-3058) - (R73-81) -43.11 67.1 94 -0.643 1.0000
## (N72-3058) - (R75-12) 154.78 67.1 94 2.308 0.4787
## (N72-3058) - Tracy -39.22 67.1 94 -0.585 1.0000
## (N72-3148) - (N73-1102) 64.56 67.1 94 0.962 0.9981
## (N72-3148) - (N73-693) 129.78 67.1 94 1.935 0.7344
## (N72-3148) - (N73-877) 162.67 67.1 94 2.425 0.4000
## (N72-3148) - (N73-882) 169.33 67.1 94 2.525 0.3381
## (N72-3148) - (R73-81) 192.22 67.1 94 2.866 0.1706
## (N72-3148) - (R75-12) 390.11 67.1 94 5.816 <.0001
## (N72-3148) - Tracy 196.11 67.1 94 2.924 0.1496
## (N73-1102) - (N73-693) 65.22 67.1 94 0.972 0.9979
## (N73-1102) - (N73-877) 98.11 67.1 94 1.463 0.9468
## (N73-1102) - (N73-882) 104.78 67.1 94 1.562 0.9181
## (N73-1102) - (R73-81) 127.67 67.1 94 1.903 0.7541
## (N73-1102) - (R75-12) 325.56 67.1 94 4.854 0.0003
## (N73-1102) - Tracy 131.56 67.1 94 1.961 0.7175
## (N73-693) - (N73-877) 32.89 67.1 94 0.490 1.0000
## (N73-693) - (N73-882) 39.56 67.1 94 0.590 1.0000
## (N73-693) - (R73-81) 62.44 67.1 94 0.931 0.9986
## (N73-693) - (R75-12) 260.33 67.1 94 3.881 0.0099
## (N73-693) - Tracy 66.33 67.1 94 0.989 0.9976
## (N73-877) - (N73-882) 6.67 67.1 94 0.099 1.0000
## (N73-877) - (R73-81) 29.56 67.1 94 0.441 1.0000
## (N73-877) - (R75-12) 227.44 67.1 94 3.391 0.0448
## (N73-877) - Tracy 33.44 67.1 94 0.499 1.0000
## (N73-882) - (R73-81) 22.89 67.1 94 0.341 1.0000
## (N73-882) - (R75-12) 220.78 67.1 94 3.292 0.0591
## (N73-882) - Tracy 26.78 67.1 94 0.399 1.0000
## (R73-81) - (R75-12) 197.89 67.1 94 2.950 0.1407
## (R73-81) - Tracy 3.89 67.1 94 0.058 1.0000
## (R75-12) - Tracy -194.00 67.1 94 -2.892 0.1608
##
## Results are averaged over the levels of: loc
## P value adjustment: tukey method for comparing a family of 12 estimates
##
## $contrasts
## contrast estimate SE df
## (Centennial - (D74-7741)) - (Centennial - (N72-137)) 77.444 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N72-3058)) -75.778 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N72-3148)) 159.556 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N73-1102)) 95.000 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N73-693)) 29.778 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N73-877)) -3.111 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (N73-882)) -9.778 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (R73-81)) -32.667 67.1 94
## (Centennial - (D74-7741)) - (Centennial - (R75-12)) -230.556 67.1 94
## (Centennial - (D74-7741)) - (Centennial - Tracy) -36.556 67.1 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N72-137)) 65.778 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N72-3058)) -87.444 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N72-3148)) 147.889 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N73-1102)) 83.333 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N73-693)) 18.111 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N73-877)) -14.778 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (N73-882)) -21.444 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (R73-81)) -44.333 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - (R75-12)) -242.222 116.2 94
## (Centennial - (D74-7741)) - ((D74-7741) - Tracy) -48.222 116.2 94
## (Centennial - (D74-7741)) - ((N72-137) - (N72-3058)) -164.889 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (N72-3148)) 70.444 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (N73-1102)) 5.889 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (N73-693)) -59.333 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (N73-877)) -92.222 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (N73-882)) -98.889 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (R73-81)) -121.778 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - (R75-12)) -319.667 94.9 94
## (Centennial - (D74-7741)) - ((N72-137) - Tracy) -125.667 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (N72-3148)) 223.667 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (N73-1102)) 159.111 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (N73-693)) 93.889 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (N73-877)) 61.000 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (N73-882)) 54.333 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (R73-81)) 31.444 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - (R75-12)) -166.444 94.9 94
## (Centennial - (D74-7741)) - ((N72-3058) - Tracy) 27.556 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (N73-1102)) -76.222 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (N73-693)) -141.444 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (N73-877)) -174.333 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (N73-882)) -181.000 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (R73-81)) -203.889 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - (R75-12)) -401.778 94.9 94
## (Centennial - (D74-7741)) - ((N72-3148) - Tracy) -207.778 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - (N73-693)) -76.889 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - (N73-877)) -109.778 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - (N73-882)) -116.444 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - (R73-81)) -139.333 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - (R75-12)) -337.222 94.9 94
## (Centennial - (D74-7741)) - ((N73-1102) - Tracy) -143.222 94.9 94
## (Centennial - (D74-7741)) - ((N73-693) - (N73-877)) -44.556 94.9 94
## (Centennial - (D74-7741)) - ((N73-693) - (N73-882)) -51.222 94.9 94
## (Centennial - (D74-7741)) - ((N73-693) - (R73-81)) -74.111 94.9 94
## (Centennial - (D74-7741)) - ((N73-693) - (R75-12)) -272.000 94.9 94
## (Centennial - (D74-7741)) - ((N73-693) - Tracy) -78.000 94.9 94
## (Centennial - (D74-7741)) - ((N73-877) - (N73-882)) -18.333 94.9 94
## (Centennial - (D74-7741)) - ((N73-877) - (R73-81)) -41.222 94.9 94
## (Centennial - (D74-7741)) - ((N73-877) - (R75-12)) -239.111 94.9 94
## (Centennial - (D74-7741)) - ((N73-877) - Tracy) -45.111 94.9 94
## (Centennial - (D74-7741)) - ((N73-882) - (R73-81)) -34.556 94.9 94
## (Centennial - (D74-7741)) - ((N73-882) - (R75-12)) -232.444 94.9 94
## (Centennial - (D74-7741)) - ((N73-882) - Tracy) -38.444 94.9 94
## (Centennial - (D74-7741)) - ((R73-81) - (R75-12)) -209.556 94.9 94
## (Centennial - (D74-7741)) - ((R73-81) - Tracy) -15.556 94.9 94
## (Centennial - (D74-7741)) - ((R75-12) - Tracy) 182.333 94.9 94
## (Centennial - (N72-137)) - (Centennial - (N72-3058)) -153.222 67.1 94
## (Centennial - (N72-137)) - (Centennial - (N72-3148)) 82.111 67.1 94
## (Centennial - (N72-137)) - (Centennial - (N73-1102)) 17.556 67.1 94
## (Centennial - (N72-137)) - (Centennial - (N73-693)) -47.667 67.1 94
## (Centennial - (N72-137)) - (Centennial - (N73-877)) -80.556 67.1 94
## (Centennial - (N72-137)) - (Centennial - (N73-882)) -87.222 67.1 94
## (Centennial - (N72-137)) - (Centennial - (R73-81)) -110.111 67.1 94
## (Centennial - (N72-137)) - (Centennial - (R75-12)) -308.000 67.1 94
## (Centennial - (N72-137)) - (Centennial - Tracy) -114.000 67.1 94
## (Centennial - (N72-137)) - ((D74-7741) - (N72-137)) -11.667 67.1 94
## (Centennial - (N72-137)) - ((D74-7741) - (N72-3058)) -164.889 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (N72-3148)) 70.444 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (N73-1102)) 5.889 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (N73-693)) -59.333 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (N73-877)) -92.222 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (N73-882)) -98.889 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (R73-81)) -121.778 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - (R75-12)) -319.667 94.9 94
## (Centennial - (N72-137)) - ((D74-7741) - Tracy) -125.667 94.9 94
## (Centennial - (N72-137)) - ((N72-137) - (N72-3058)) -242.333 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (N72-3148)) -7.000 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (N73-1102)) -71.556 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (N73-693)) -136.778 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (N73-877)) -169.667 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (N73-882)) -176.333 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (R73-81)) -199.222 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - (R75-12)) -397.111 116.2 94
## (Centennial - (N72-137)) - ((N72-137) - Tracy) -203.111 116.2 94
## (Centennial - (N72-137)) - ((N72-3058) - (N72-3148)) 146.222 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (N73-1102)) 81.667 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (N73-693)) 16.444 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (N73-877)) -16.444 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (N73-882)) -23.111 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (R73-81)) -46.000 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - (R75-12)) -243.889 94.9 94
## (Centennial - (N72-137)) - ((N72-3058) - Tracy) -49.889 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (N73-1102)) -153.667 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (N73-693)) -218.889 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (N73-877)) -251.778 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (N73-882)) -258.444 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (R73-81)) -281.333 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - (R75-12)) -479.222 94.9 94
## (Centennial - (N72-137)) - ((N72-3148) - Tracy) -285.222 94.9 94
## (Centennial - (N72-137)) - ((N73-1102) - (N73-693)) -154.333 94.9 94
## (Centennial - (N72-137)) - ((N73-1102) - (N73-877)) -187.222 94.9 94
## (Centennial - (N72-137)) - ((N73-1102) - (N73-882)) -193.889 94.9 94
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## (Centennial - (N72-137)) - ((N73-1102) - (R75-12)) -414.667 94.9 94
## (Centennial - (N72-137)) - ((N73-1102) - Tracy) -220.667 94.9 94
## (Centennial - (N72-137)) - ((N73-693) - (N73-877)) -122.000 94.9 94
## (Centennial - (N72-137)) - ((N73-693) - (N73-882)) -128.667 94.9 94
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## (Centennial - (N72-137)) - ((N73-693) - (R75-12)) -349.444 94.9 94
## (Centennial - (N72-137)) - ((N73-693) - Tracy) -155.444 94.9 94
## (Centennial - (N72-137)) - ((N73-877) - (N73-882)) -95.778 94.9 94
## (Centennial - (N72-137)) - ((N73-877) - (R73-81)) -118.667 94.9 94
## (Centennial - (N72-137)) - ((N73-877) - (R75-12)) -316.556 94.9 94
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## (Centennial - (N72-137)) - ((N73-882) - (R73-81)) -112.000 94.9 94
## (Centennial - (N72-137)) - ((N73-882) - (R75-12)) -309.889 94.9 94
## (Centennial - (N72-137)) - ((N73-882) - Tracy) -115.889 94.9 94
## (Centennial - (N72-137)) - ((R73-81) - (R75-12)) -287.000 94.9 94
## (Centennial - (N72-137)) - ((R73-81) - Tracy) -93.000 94.9 94
## (Centennial - (N72-137)) - ((R75-12) - Tracy) 104.889 94.9 94
## (Centennial - (N72-3058)) - (Centennial - (N72-3148)) 235.333 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (N73-1102)) 170.778 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (N73-693)) 105.556 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (N73-877)) 72.667 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (N73-882)) 66.000 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (R73-81)) 43.111 67.1 94
## (Centennial - (N72-3058)) - (Centennial - (R75-12)) -154.778 67.1 94
## (Centennial - (N72-3058)) - (Centennial - Tracy) 39.222 67.1 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N72-137)) 141.556 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N72-3058)) -11.667 67.1 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N72-3148)) 223.667 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N73-1102)) 159.111 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N73-693)) 93.889 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N73-877)) 61.000 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (N73-882)) 54.333 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (R73-81)) 31.444 94.9 94
## (Centennial - (N72-3058)) - ((D74-7741) - (R75-12)) -166.444 94.9 94
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## (Centennial - (N72-3058)) - ((N72-137) - (N72-3058)) -89.111 67.1 94
## (Centennial - (N72-3058)) - ((N72-137) - (N72-3148)) 146.222 94.9 94
## (Centennial - (N72-3058)) - ((N72-137) - (N73-1102)) 81.667 94.9 94
## (Centennial - (N72-3058)) - ((N72-137) - (N73-693)) 16.444 94.9 94
## (Centennial - (N72-3058)) - ((N72-137) - (N73-877)) -16.444 94.9 94
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## (Centennial - (N72-3058)) - ((N72-137) - Tracy) -49.889 94.9 94
## (Centennial - (N72-3058)) - ((N72-3058) - (N72-3148)) 299.444 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (N73-1102)) 234.889 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (N73-693)) 169.667 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (N73-877)) 136.778 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (N73-882)) 130.111 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (R73-81)) 107.222 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - (R75-12)) -90.667 116.2 94
## (Centennial - (N72-3058)) - ((N72-3058) - Tracy) 103.333 116.2 94
## (Centennial - (N72-3058)) - ((N72-3148) - (N73-1102)) -0.444 94.9 94
## (Centennial - (N72-3058)) - ((N72-3148) - (N73-693)) -65.667 94.9 94
## (Centennial - (N72-3058)) - ((N72-3148) - (N73-877)) -98.556 94.9 94
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## (Centennial - (N72-3058)) - ((N72-3148) - (R75-12)) -326.000 94.9 94
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## (Centennial - (N72-3058)) - ((N73-1102) - (N73-877)) -34.000 94.9 94
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## (Centennial - (N72-3058)) - ((N73-1102) - (R75-12)) -261.444 94.9 94
## (Centennial - (N72-3058)) - ((N73-1102) - Tracy) -67.444 94.9 94
## (Centennial - (N72-3058)) - ((N73-693) - (N73-877)) 31.222 94.9 94
## (Centennial - (N72-3058)) - ((N73-693) - (N73-882)) 24.556 94.9 94
## (Centennial - (N72-3058)) - ((N73-693) - (R73-81)) 1.667 94.9 94
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## (Centennial - (N72-3058)) - ((N73-877) - (N73-882)) 57.444 94.9 94
## (Centennial - (N72-3058)) - ((N73-877) - (R73-81)) 34.556 94.9 94
## (Centennial - (N72-3058)) - ((N73-877) - (R75-12)) -163.333 94.9 94
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## (Centennial - (N72-3058)) - ((N73-882) - (R75-12)) -156.667 94.9 94
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## (Centennial - (N72-3148)) - (Centennial - (N73-693)) -129.778 67.1 94
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## (Centennial - (N72-3148)) - (Centennial - (R75-12)) -390.111 67.1 94
## (Centennial - (N72-3148)) - (Centennial - Tracy) -196.111 67.1 94
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## (Centennial - (N72-3148)) - ((D74-7741) - (R75-12)) -401.778 94.9 94
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## (Centennial - (N72-3148)) - ((N72-3058) - (N72-3148)) 64.111 67.1 94
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## (Centennial - (N72-3148)) - ((N72-3148) - (R75-12)) -561.333 116.2 94
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## (Centennial - (N73-693)) - ((N72-3058) - (N73-1102)) 129.333 94.9 94
## (Centennial - (N73-693)) - ((N72-3058) - (N73-693)) 64.111 67.1 94
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## (Centennial - (N73-882)) - ((D74-7741) - (N72-3058)) -77.667 94.9 94
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## (Centennial - (R73-81)) - ((N72-3058) - (R75-12)) -133.778 94.9 94
## (Centennial - (R73-81)) - ((N72-3058) - Tracy) 60.222 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - (N73-1102)) -43.556 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - (N73-693)) -108.778 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - (N73-877)) -141.667 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - (N73-882)) -148.333 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - (R73-81)) -171.222 67.1 94
## (Centennial - (R73-81)) - ((N72-3148) - (R75-12)) -369.111 94.9 94
## (Centennial - (R73-81)) - ((N72-3148) - Tracy) -175.111 94.9 94
## (Centennial - (R73-81)) - ((N73-1102) - (N73-693)) -44.222 94.9 94
## (Centennial - (R73-81)) - ((N73-1102) - (N73-877)) -77.111 94.9 94
## (Centennial - (R73-81)) - ((N73-1102) - (N73-882)) -83.778 94.9 94
## (Centennial - (R73-81)) - ((N73-1102) - (R73-81)) -106.667 67.1 94
## (Centennial - (R73-81)) - ((N73-1102) - (R75-12)) -304.556 94.9 94
## (Centennial - (R73-81)) - ((N73-1102) - Tracy) -110.556 94.9 94
## (Centennial - (R73-81)) - ((N73-693) - (N73-877)) -11.889 94.9 94
## (Centennial - (R73-81)) - ((N73-693) - (N73-882)) -18.556 94.9 94
## (Centennial - (R73-81)) - ((N73-693) - (R73-81)) -41.444 67.1 94
## (Centennial - (R73-81)) - ((N73-693) - (R75-12)) -239.333 94.9 94
## (Centennial - (R73-81)) - ((N73-693) - Tracy) -45.333 94.9 94
## (Centennial - (R73-81)) - ((N73-877) - (N73-882)) 14.333 94.9 94
## (Centennial - (R73-81)) - ((N73-877) - (R73-81)) -8.556 67.1 94
## (Centennial - (R73-81)) - ((N73-877) - (R75-12)) -206.444 94.9 94
## (Centennial - (R73-81)) - ((N73-877) - Tracy) -12.444 94.9 94
## (Centennial - (R73-81)) - ((N73-882) - (R73-81)) -1.889 67.1 94
## (Centennial - (R73-81)) - ((N73-882) - (R75-12)) -199.778 94.9 94
## (Centennial - (R73-81)) - ((N73-882) - Tracy) -5.778 94.9 94
## (Centennial - (R73-81)) - ((R73-81) - (R75-12)) -176.889 116.2 94
## (Centennial - (R73-81)) - ((R73-81) - Tracy) 17.111 116.2 94
## (Centennial - (R73-81)) - ((R75-12) - Tracy) 215.000 94.9 94
## (Centennial - (R75-12)) - (Centennial - Tracy) 194.000 67.1 94
## (Centennial - (R75-12)) - ((D74-7741) - (N72-137)) 296.333 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N72-3058)) 143.111 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N72-3148)) 378.444 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N73-1102)) 313.889 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N73-693)) 248.667 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N73-877)) 215.778 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (N73-882)) 209.111 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (R73-81)) 186.222 94.9 94
## (Centennial - (R75-12)) - ((D74-7741) - (R75-12)) -11.667 67.1 94
## (Centennial - (R75-12)) - ((D74-7741) - Tracy) 182.333 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N72-3058)) 65.667 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N72-3148)) 301.000 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N73-1102)) 236.444 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N73-693)) 171.222 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N73-877)) 138.333 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (N73-882)) 131.667 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (R73-81)) 108.778 94.9 94
## (Centennial - (R75-12)) - ((N72-137) - (R75-12)) -89.111 67.1 94
## (Centennial - (R75-12)) - ((N72-137) - Tracy) 104.889 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (N72-3148)) 454.222 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (N73-1102)) 389.667 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (N73-693)) 324.444 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (N73-877)) 291.556 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (N73-882)) 284.889 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (R73-81)) 262.000 94.9 94
## (Centennial - (R75-12)) - ((N72-3058) - (R75-12)) 64.111 67.1 94
## (Centennial - (R75-12)) - ((N72-3058) - Tracy) 258.111 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (N73-1102)) 154.333 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (N73-693)) 89.111 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (N73-877)) 56.222 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (N73-882)) 49.556 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (R73-81)) 26.667 94.9 94
## (Centennial - (R75-12)) - ((N72-3148) - (R75-12)) -171.222 67.1 94
## (Centennial - (R75-12)) - ((N72-3148) - Tracy) 22.778 94.9 94
## (Centennial - (R75-12)) - ((N73-1102) - (N73-693)) 153.667 94.9 94
## (Centennial - (R75-12)) - ((N73-1102) - (N73-877)) 120.778 94.9 94
## (Centennial - (R75-12)) - ((N73-1102) - (N73-882)) 114.111 94.9 94
## (Centennial - (R75-12)) - ((N73-1102) - (R73-81)) 91.222 94.9 94
## (Centennial - (R75-12)) - ((N73-1102) - (R75-12)) -106.667 67.1 94
## (Centennial - (R75-12)) - ((N73-1102) - Tracy) 87.333 94.9 94
## (Centennial - (R75-12)) - ((N73-693) - (N73-877)) 186.000 94.9 94
## (Centennial - (R75-12)) - ((N73-693) - (N73-882)) 179.333 94.9 94
## (Centennial - (R75-12)) - ((N73-693) - (R73-81)) 156.444 94.9 94
## (Centennial - (R75-12)) - ((N73-693) - (R75-12)) -41.444 67.1 94
## (Centennial - (R75-12)) - ((N73-693) - Tracy) 152.556 94.9 94
## (Centennial - (R75-12)) - ((N73-877) - (N73-882)) 212.222 94.9 94
## (Centennial - (R75-12)) - ((N73-877) - (R73-81)) 189.333 94.9 94
## (Centennial - (R75-12)) - ((N73-877) - (R75-12)) -8.556 67.1 94
## (Centennial - (R75-12)) - ((N73-877) - Tracy) 185.444 94.9 94
## (Centennial - (R75-12)) - ((N73-882) - (R73-81)) 196.000 94.9 94
## (Centennial - (R75-12)) - ((N73-882) - (R75-12)) -1.889 67.1 94
## (Centennial - (R75-12)) - ((N73-882) - Tracy) 192.111 94.9 94
## (Centennial - (R75-12)) - ((R73-81) - (R75-12)) 21.000 67.1 94
## (Centennial - (R75-12)) - ((R73-81) - Tracy) 215.000 94.9 94
## (Centennial - (R75-12)) - ((R75-12) - Tracy) 412.889 116.2 94
## (Centennial - Tracy) - ((D74-7741) - (N72-137)) 102.333 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N72-3058)) -50.889 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N72-3148)) 184.444 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N73-1102)) 119.889 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N73-693)) 54.667 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N73-877)) 21.778 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (N73-882)) 15.111 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (R73-81)) -7.778 94.9 94
## (Centennial - Tracy) - ((D74-7741) - (R75-12)) -205.667 94.9 94
## (Centennial - Tracy) - ((D74-7741) - Tracy) -11.667 67.1 94
## (Centennial - Tracy) - ((N72-137) - (N72-3058)) -128.333 94.9 94
## (Centennial - Tracy) - ((N72-137) - (N72-3148)) 107.000 94.9 94
## (Centennial - Tracy) - ((N72-137) - (N73-1102)) 42.444 94.9 94
## (Centennial - Tracy) - ((N72-137) - (N73-693)) -22.778 94.9 94
## (Centennial - Tracy) - ((N72-137) - (N73-877)) -55.667 94.9 94
## (Centennial - Tracy) - ((N72-137) - (N73-882)) -62.333 94.9 94
## (Centennial - Tracy) - ((N72-137) - (R73-81)) -85.222 94.9 94
## (Centennial - Tracy) - ((N72-137) - (R75-12)) -283.111 94.9 94
## (Centennial - Tracy) - ((N72-137) - Tracy) -89.111 67.1 94
## (Centennial - Tracy) - ((N72-3058) - (N72-3148)) 260.222 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (N73-1102)) 195.667 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (N73-693)) 130.444 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (N73-877)) 97.556 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (N73-882)) 90.889 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (R73-81)) 68.000 94.9 94
## (Centennial - Tracy) - ((N72-3058) - (R75-12)) -129.889 94.9 94
## (Centennial - Tracy) - ((N72-3058) - Tracy) 64.111 67.1 94
## (Centennial - Tracy) - ((N72-3148) - (N73-1102)) -39.667 94.9 94
## (Centennial - Tracy) - ((N72-3148) - (N73-693)) -104.889 94.9 94
## (Centennial - Tracy) - ((N72-3148) - (N73-877)) -137.778 94.9 94
## (Centennial - Tracy) - ((N72-3148) - (N73-882)) -144.444 94.9 94
## (Centennial - Tracy) - ((N72-3148) - (R73-81)) -167.333 94.9 94
## (Centennial - Tracy) - ((N72-3148) - (R75-12)) -365.222 94.9 94
## (Centennial - Tracy) - ((N72-3148) - Tracy) -171.222 67.1 94
## (Centennial - Tracy) - ((N73-1102) - (N73-693)) -40.333 94.9 94
## (Centennial - Tracy) - ((N73-1102) - (N73-877)) -73.222 94.9 94
## (Centennial - Tracy) - ((N73-1102) - (N73-882)) -79.889 94.9 94
## (Centennial - Tracy) - ((N73-1102) - (R73-81)) -102.778 94.9 94
## (Centennial - Tracy) - ((N73-1102) - (R75-12)) -300.667 94.9 94
## (Centennial - Tracy) - ((N73-1102) - Tracy) -106.667 67.1 94
## (Centennial - Tracy) - ((N73-693) - (N73-877)) -8.000 94.9 94
## (Centennial - Tracy) - ((N73-693) - (N73-882)) -14.667 94.9 94
## (Centennial - Tracy) - ((N73-693) - (R73-81)) -37.556 94.9 94
## (Centennial - Tracy) - ((N73-693) - (R75-12)) -235.444 94.9 94
## (Centennial - Tracy) - ((N73-693) - Tracy) -41.444 67.1 94
## (Centennial - Tracy) - ((N73-877) - (N73-882)) 18.222 94.9 94
## (Centennial - Tracy) - ((N73-877) - (R73-81)) -4.667 94.9 94
## (Centennial - Tracy) - ((N73-877) - (R75-12)) -202.556 94.9 94
## (Centennial - Tracy) - ((N73-877) - Tracy) -8.556 67.1 94
## (Centennial - Tracy) - ((N73-882) - (R73-81)) 2.000 94.9 94
## (Centennial - Tracy) - ((N73-882) - (R75-12)) -195.889 94.9 94
## (Centennial - Tracy) - ((N73-882) - Tracy) -1.889 67.1 94
## (Centennial - Tracy) - ((R73-81) - (R75-12)) -173.000 94.9 94
## (Centennial - Tracy) - ((R73-81) - Tracy) 21.000 67.1 94
## (Centennial - Tracy) - ((R75-12) - Tracy) 218.889 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N72-3058)) -153.222 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N72-3148)) 82.111 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-1102)) 17.556 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-693)) -47.667 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-877)) -80.556 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-882)) -87.222 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (R73-81)) -110.111 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - (R75-12)) -308.000 67.1 94
## ((D74-7741) - (N72-137)) - ((D74-7741) - Tracy) -114.000 67.1 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N72-3058)) -230.667 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N72-3148)) 4.667 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N73-1102)) -59.889 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N73-693)) -125.111 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N73-877)) -158.000 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (N73-882)) -164.667 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (R73-81)) -187.556 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - (R75-12)) -385.444 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-137) - Tracy) -191.444 116.2 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (N72-3148)) 157.889 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-1102)) 93.333 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-693)) 28.111 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-877)) -4.778 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-882)) -11.444 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (R73-81)) -34.333 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - (R75-12)) -232.222 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3058) - Tracy) -38.222 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-1102)) -142.000 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-693)) -207.222 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-877)) -240.111 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-882)) -246.778 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (R73-81)) -269.667 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - (R75-12)) -467.556 94.9 94
## ((D74-7741) - (N72-137)) - ((N72-3148) - Tracy) -273.556 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-693)) -142.667 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-877)) -175.556 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-882)) -182.222 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - (R73-81)) -205.111 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - (R75-12)) -403.000 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-1102) - Tracy) -209.000 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-693) - (N73-877)) -110.333 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-693) - (N73-882)) -117.000 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-693) - (R73-81)) -139.889 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-693) - (R75-12)) -337.778 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-693) - Tracy) -143.778 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-877) - (N73-882)) -84.111 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-877) - (R73-81)) -107.000 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-877) - (R75-12)) -304.889 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-877) - Tracy) -110.889 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-882) - (R73-81)) -100.333 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-882) - (R75-12)) -298.222 94.9 94
## ((D74-7741) - (N72-137)) - ((N73-882) - Tracy) -104.222 94.9 94
## ((D74-7741) - (N72-137)) - ((R73-81) - (R75-12)) -275.333 94.9 94
## ((D74-7741) - (N72-137)) - ((R73-81) - Tracy) -81.333 94.9 94
## ((D74-7741) - (N72-137)) - ((R75-12) - Tracy) 116.556 94.9 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (N72-3148)) 235.333 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-1102)) 170.778 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-693)) 105.556 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-877)) 72.667 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-882)) 66.000 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (R73-81)) 43.111 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - (R75-12)) -154.778 67.1 94
## ((D74-7741) - (N72-3058)) - ((D74-7741) - Tracy) 39.222 67.1 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N72-3058)) -77.444 67.1 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N72-3148)) 157.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-1102)) 93.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-693)) 28.111 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-877)) -4.778 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-882)) -11.444 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (R73-81)) -34.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - (R75-12)) -232.222 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-137) - Tracy) -38.222 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (N72-3148)) 311.111 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-1102)) 246.556 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-693)) 181.333 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-877)) 148.444 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-882)) 141.778 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (R73-81)) 118.889 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - (R75-12)) -79.000 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3058) - Tracy) 115.000 116.2 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-1102)) 11.222 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-693)) -54.000 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-877)) -86.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-882)) -93.556 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (R73-81)) -116.444 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - (R75-12)) -314.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N72-3148) - Tracy) -120.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - (N73-693)) 10.556 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - (N73-877)) -22.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - (N73-882)) -29.000 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - (R73-81)) -51.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - (R75-12)) -249.778 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-1102) - Tracy) -55.778 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-693) - (N73-877)) 42.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-693) - (N73-882)) 36.222 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-693) - (R73-81)) 13.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-693) - (R75-12)) -184.556 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-693) - Tracy) 9.444 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-877) - (N73-882)) 69.111 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-877) - (R73-81)) 46.222 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-877) - (R75-12)) -151.667 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-877) - Tracy) 42.333 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-882) - (R73-81)) 52.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-882) - (R75-12)) -145.000 94.9 94
## ((D74-7741) - (N72-3058)) - ((N73-882) - Tracy) 49.000 94.9 94
## ((D74-7741) - (N72-3058)) - ((R73-81) - (R75-12)) -122.111 94.9 94
## ((D74-7741) - (N72-3058)) - ((R73-81) - Tracy) 71.889 94.9 94
## ((D74-7741) - (N72-3058)) - ((R75-12) - Tracy) 269.778 94.9 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (N73-1102)) -64.556 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (N73-693)) -129.778 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (N73-877)) -162.667 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (N73-882)) -169.333 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (R73-81)) -192.222 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - (R75-12)) -390.111 67.1 94
## ((D74-7741) - (N72-3148)) - ((D74-7741) - Tracy) -196.111 67.1 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N72-3058)) -312.778 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N72-3148)) -77.444 67.1 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N73-1102)) -142.000 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N73-693)) -207.222 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N73-877)) -240.111 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (N73-882)) -246.778 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (R73-81)) -269.667 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - (R75-12)) -467.556 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-137) - Tracy) -273.556 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (N72-3148)) 75.778 67.1 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (N73-1102)) 11.222 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (N73-693)) -54.000 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (N73-877)) -86.889 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (N73-882)) -93.556 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (R73-81)) -116.444 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - (R75-12)) -314.333 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3058) - Tracy) -120.333 94.9 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (N73-1102)) -224.111 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (N73-693)) -289.333 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (N73-877)) -322.222 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (N73-882)) -328.889 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (R73-81)) -351.778 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - (R75-12)) -549.667 116.2 94
## ((D74-7741) - (N72-3148)) - ((N72-3148) - Tracy) -355.667 116.2 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - (N73-693)) -224.778 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - (N73-877)) -257.667 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - (N73-882)) -264.333 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - (R73-81)) -287.222 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - (R75-12)) -485.111 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-1102) - Tracy) -291.111 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-693) - (N73-877)) -192.444 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-693) - (N73-882)) -199.111 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-693) - (R73-81)) -222.000 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-693) - (R75-12)) -419.889 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-693) - Tracy) -225.889 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-877) - (N73-882)) -166.222 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-877) - (R73-81)) -189.111 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-877) - (R75-12)) -387.000 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-877) - Tracy) -193.000 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-882) - (R73-81)) -182.444 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-882) - (R75-12)) -380.333 94.9 94
## ((D74-7741) - (N72-3148)) - ((N73-882) - Tracy) -186.333 94.9 94
## ((D74-7741) - (N72-3148)) - ((R73-81) - (R75-12)) -357.444 94.9 94
## ((D74-7741) - (N72-3148)) - ((R73-81) - Tracy) -163.444 94.9 94
## ((D74-7741) - (N72-3148)) - ((R75-12) - Tracy) 34.444 94.9 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - (N73-693)) -65.222 67.1 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - (N73-877)) -98.111 67.1 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - (N73-882)) -104.778 67.1 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - (R73-81)) -127.667 67.1 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - (R75-12)) -325.556 67.1 94
## ((D74-7741) - (N73-1102)) - ((D74-7741) - Tracy) -131.556 67.1 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N72-3058)) -248.222 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N72-3148)) -12.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N73-1102)) -77.444 67.1 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N73-693)) -142.667 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N73-877)) -175.556 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (N73-882)) -182.222 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (R73-81)) -205.111 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - (R75-12)) -403.000 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-137) - Tracy) -209.000 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (N72-3148)) 140.333 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (N73-1102)) 75.778 67.1 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (N73-693)) 10.556 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (N73-877)) -22.333 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (N73-882)) -29.000 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (R73-81)) -51.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - (R75-12)) -249.778 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3058) - Tracy) -55.778 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (N73-1102)) -159.556 67.1 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (N73-693)) -224.778 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (N73-877)) -257.667 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (N73-882)) -264.333 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (R73-81)) -287.222 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - (R75-12)) -485.111 94.9 94
## ((D74-7741) - (N73-1102)) - ((N72-3148) - Tracy) -291.111 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - (N73-693)) -160.222 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - (N73-877)) -193.111 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - (N73-882)) -199.778 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - (R73-81)) -222.667 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - (R75-12)) -420.556 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-1102) - Tracy) -226.556 116.2 94
## ((D74-7741) - (N73-1102)) - ((N73-693) - (N73-877)) -127.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-693) - (N73-882)) -134.556 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-693) - (R73-81)) -157.444 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-693) - (R75-12)) -355.333 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-693) - Tracy) -161.333 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-877) - (N73-882)) -101.667 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-877) - (R73-81)) -124.556 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-877) - (R75-12)) -322.444 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-877) - Tracy) -128.444 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-882) - (R73-81)) -117.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-882) - (R75-12)) -315.778 94.9 94
## ((D74-7741) - (N73-1102)) - ((N73-882) - Tracy) -121.778 94.9 94
## ((D74-7741) - (N73-1102)) - ((R73-81) - (R75-12)) -292.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((R73-81) - Tracy) -98.889 94.9 94
## ((D74-7741) - (N73-1102)) - ((R75-12) - Tracy) 99.000 94.9 94
## ((D74-7741) - (N73-693)) - ((D74-7741) - (N73-877)) -32.889 67.1 94
## ((D74-7741) - (N73-693)) - ((D74-7741) - (N73-882)) -39.556 67.1 94
## ((D74-7741) - (N73-693)) - ((D74-7741) - (R73-81)) -62.444 67.1 94
## ((D74-7741) - (N73-693)) - ((D74-7741) - (R75-12)) -260.333 67.1 94
## ((D74-7741) - (N73-693)) - ((D74-7741) - Tracy) -66.333 67.1 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N72-3058)) -183.000 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N72-3148)) 52.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N73-1102)) -12.222 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N73-693)) -77.444 67.1 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N73-877)) -110.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (N73-882)) -117.000 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (R73-81)) -139.889 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - (R75-12)) -337.778 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-137) - Tracy) -143.778 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (N72-3148)) 205.556 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (N73-1102)) 141.000 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (N73-693)) 75.778 67.1 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (N73-877)) 42.889 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (N73-882)) 36.222 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (R73-81)) 13.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - (R75-12)) -184.556 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3058) - Tracy) 9.444 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (N73-1102)) -94.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (N73-693)) -159.556 67.1 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (N73-877)) -192.444 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (N73-882)) -199.111 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (R73-81)) -222.000 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - (R75-12)) -419.889 94.9 94
## ((D74-7741) - (N73-693)) - ((N72-3148) - Tracy) -225.889 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - (N73-693)) -95.000 67.1 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - (N73-877)) -127.889 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - (N73-882)) -134.556 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - (R73-81)) -157.444 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - (R75-12)) -355.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-1102) - Tracy) -161.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-693) - (N73-877)) -62.667 116.2 94
## ((D74-7741) - (N73-693)) - ((N73-693) - (N73-882)) -69.333 116.2 94
## ((D74-7741) - (N73-693)) - ((N73-693) - (R73-81)) -92.222 116.2 94
## ((D74-7741) - (N73-693)) - ((N73-693) - (R75-12)) -290.111 116.2 94
## ((D74-7741) - (N73-693)) - ((N73-693) - Tracy) -96.111 116.2 94
## ((D74-7741) - (N73-693)) - ((N73-877) - (N73-882)) -36.444 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-877) - (R73-81)) -59.333 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-877) - (R75-12)) -257.222 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-877) - Tracy) -63.222 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-882) - (R73-81)) -52.667 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-882) - (R75-12)) -250.556 94.9 94
## ((D74-7741) - (N73-693)) - ((N73-882) - Tracy) -56.556 94.9 94
## ((D74-7741) - (N73-693)) - ((R73-81) - (R75-12)) -227.667 94.9 94
## ((D74-7741) - (N73-693)) - ((R73-81) - Tracy) -33.667 94.9 94
## ((D74-7741) - (N73-693)) - ((R75-12) - Tracy) 164.222 94.9 94
## ((D74-7741) - (N73-877)) - ((D74-7741) - (N73-882)) -6.667 67.1 94
## ((D74-7741) - (N73-877)) - ((D74-7741) - (R73-81)) -29.556 67.1 94
## ((D74-7741) - (N73-877)) - ((D74-7741) - (R75-12)) -227.444 67.1 94
## ((D74-7741) - (N73-877)) - ((D74-7741) - Tracy) -33.444 67.1 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N72-3058)) -150.111 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N72-3148)) 85.222 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N73-1102)) 20.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N73-693)) -44.556 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N73-877)) -77.444 67.1 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (N73-882)) -84.111 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (R73-81)) -107.000 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - (R75-12)) -304.889 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-137) - Tracy) -110.889 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (N72-3148)) 238.444 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (N73-1102)) 173.889 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (N73-693)) 108.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (N73-877)) 75.778 67.1 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (N73-882)) 69.111 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (R73-81)) 46.222 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - (R75-12)) -151.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3058) - Tracy) 42.333 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (N73-1102)) -61.444 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (N73-693)) -126.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (N73-877)) -159.556 67.1 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (N73-882)) -166.222 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (R73-81)) -189.111 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - (R75-12)) -387.000 94.9 94
## ((D74-7741) - (N73-877)) - ((N72-3148) - Tracy) -193.000 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - (N73-693)) -62.111 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - (N73-877)) -95.000 67.1 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - (N73-882)) -101.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - (R73-81)) -124.556 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - (R75-12)) -322.444 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-1102) - Tracy) -128.444 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-693) - (N73-877)) -29.778 67.1 94
## ((D74-7741) - (N73-877)) - ((N73-693) - (N73-882)) -36.444 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-693) - (R73-81)) -59.333 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-693) - (R75-12)) -257.222 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-693) - Tracy) -63.222 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-877) - (N73-882)) -3.556 116.2 94
## ((D74-7741) - (N73-877)) - ((N73-877) - (R73-81)) -26.444 116.2 94
## ((D74-7741) - (N73-877)) - ((N73-877) - (R75-12)) -224.333 116.2 94
## ((D74-7741) - (N73-877)) - ((N73-877) - Tracy) -30.333 116.2 94
## ((D74-7741) - (N73-877)) - ((N73-882) - (R73-81)) -19.778 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-882) - (R75-12)) -217.667 94.9 94
## ((D74-7741) - (N73-877)) - ((N73-882) - Tracy) -23.667 94.9 94
## ((D74-7741) - (N73-877)) - ((R73-81) - (R75-12)) -194.778 94.9 94
## ((D74-7741) - (N73-877)) - ((R73-81) - Tracy) -0.778 94.9 94
## ((D74-7741) - (N73-877)) - ((R75-12) - Tracy) 197.111 94.9 94
## ((D74-7741) - (N73-882)) - ((D74-7741) - (R73-81)) -22.889 67.1 94
## ((D74-7741) - (N73-882)) - ((D74-7741) - (R75-12)) -220.778 67.1 94
## ((D74-7741) - (N73-882)) - ((D74-7741) - Tracy) -26.778 67.1 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N72-3058)) -143.444 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N72-3148)) 91.889 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N73-1102)) 27.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N73-693)) -37.889 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N73-877)) -70.778 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (N73-882)) -77.444 67.1 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (R73-81)) -100.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - (R75-12)) -298.222 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-137) - Tracy) -104.222 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (N72-3148)) 245.111 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-1102)) 180.556 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-693)) 115.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-877)) 82.444 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-882)) 75.778 67.1 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (R73-81)) 52.889 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - (R75-12)) -145.000 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3058) - Tracy) 49.000 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (N73-1102)) -54.778 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (N73-693)) -120.000 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (N73-877)) -152.889 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (N73-882)) -159.556 67.1 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (R73-81)) -182.444 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - (R75-12)) -380.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N72-3148) - Tracy) -186.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - (N73-693)) -55.444 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - (N73-877)) -88.333 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - (N73-882)) -95.000 67.1 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - (R73-81)) -117.889 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - (R75-12)) -315.778 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-1102) - Tracy) -121.778 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-693) - (N73-877)) -23.111 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-693) - (N73-882)) -29.778 67.1 94
## ((D74-7741) - (N73-882)) - ((N73-693) - (R73-81)) -52.667 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-693) - (R75-12)) -250.556 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-693) - Tracy) -56.556 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-877) - (N73-882)) 3.111 67.1 94
## ((D74-7741) - (N73-882)) - ((N73-877) - (R73-81)) -19.778 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-877) - (R75-12)) -217.667 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-877) - Tracy) -23.667 94.9 94
## ((D74-7741) - (N73-882)) - ((N73-882) - (R73-81)) -13.111 116.2 94
## ((D74-7741) - (N73-882)) - ((N73-882) - (R75-12)) -211.000 116.2 94
## ((D74-7741) - (N73-882)) - ((N73-882) - Tracy) -17.000 116.2 94
## ((D74-7741) - (N73-882)) - ((R73-81) - (R75-12)) -188.111 94.9 94
## ((D74-7741) - (N73-882)) - ((R73-81) - Tracy) 5.889 94.9 94
## ((D74-7741) - (N73-882)) - ((R75-12) - Tracy) 203.778 94.9 94
## ((D74-7741) - (R73-81)) - ((D74-7741) - (R75-12)) -197.889 67.1 94
## ((D74-7741) - (R73-81)) - ((D74-7741) - Tracy) -3.889 67.1 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N72-3058)) -120.556 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N72-3148)) 114.778 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N73-1102)) 50.222 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N73-693)) -15.000 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N73-877)) -47.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (N73-882)) -54.556 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (R73-81)) -77.444 67.1 94
## ((D74-7741) - (R73-81)) - ((N72-137) - (R75-12)) -275.333 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-137) - Tracy) -81.333 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (N72-3148)) 268.000 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-1102)) 203.444 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-693)) 138.222 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-877)) 105.333 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-882)) 98.667 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (R73-81)) 75.778 67.1 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - (R75-12)) -122.111 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3058) - Tracy) 71.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (N73-1102)) -31.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (N73-693)) -97.111 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (N73-877)) -130.000 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (N73-882)) -136.667 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (R73-81)) -159.556 67.1 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - (R75-12)) -357.444 94.9 94
## ((D74-7741) - (R73-81)) - ((N72-3148) - Tracy) -163.444 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - (N73-693)) -32.556 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - (N73-877)) -65.444 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - (N73-882)) -72.111 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - (R73-81)) -95.000 67.1 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - (R75-12)) -292.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-1102) - Tracy) -98.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-693) - (N73-877)) -0.222 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-693) - (N73-882)) -6.889 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-693) - (R73-81)) -29.778 67.1 94
## ((D74-7741) - (R73-81)) - ((N73-693) - (R75-12)) -227.667 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-693) - Tracy) -33.667 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-877) - (N73-882)) 26.000 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-877) - (R73-81)) 3.111 67.1 94
## ((D74-7741) - (R73-81)) - ((N73-877) - (R75-12)) -194.778 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-877) - Tracy) -0.778 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-882) - (R73-81)) 9.778 67.1 94
## ((D74-7741) - (R73-81)) - ((N73-882) - (R75-12)) -188.111 94.9 94
## ((D74-7741) - (R73-81)) - ((N73-882) - Tracy) 5.889 94.9 94
## ((D74-7741) - (R73-81)) - ((R73-81) - (R75-12)) -165.222 116.2 94
## ((D74-7741) - (R73-81)) - ((R73-81) - Tracy) 28.778 116.2 94
## ((D74-7741) - (R73-81)) - ((R75-12) - Tracy) 226.667 94.9 94
## ((D74-7741) - (R75-12)) - ((D74-7741) - Tracy) 194.000 67.1 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N72-3058)) 77.333 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N72-3148)) 312.667 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N73-1102)) 248.111 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N73-693)) 182.889 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N73-877)) 150.000 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (N73-882)) 143.333 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (R73-81)) 120.444 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-137) - (R75-12)) -77.444 67.1 94
## ((D74-7741) - (R75-12)) - ((N72-137) - Tracy) 116.556 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (N72-3148)) 465.889 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-1102)) 401.333 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-693)) 336.111 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-877)) 303.222 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-882)) 296.556 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (R73-81)) 273.667 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - (R75-12)) 75.778 67.1 94
## ((D74-7741) - (R75-12)) - ((N72-3058) - Tracy) 269.778 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-1102)) 166.000 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-693)) 100.778 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-877)) 67.889 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-882)) 61.222 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (R73-81)) 38.333 94.9 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - (R75-12)) -159.556 67.1 94
## ((D74-7741) - (R75-12)) - ((N72-3148) - Tracy) 34.444 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-693)) 165.333 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-877)) 132.444 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-882)) 125.778 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - (R73-81)) 102.889 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - (R75-12)) -95.000 67.1 94
## ((D74-7741) - (R75-12)) - ((N73-1102) - Tracy) 99.000 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-693) - (N73-877)) 197.667 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-693) - (N73-882)) 191.000 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-693) - (R73-81)) 168.111 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-693) - (R75-12)) -29.778 67.1 94
## ((D74-7741) - (R75-12)) - ((N73-693) - Tracy) 164.222 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-877) - (N73-882)) 223.889 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-877) - (R73-81)) 201.000 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-877) - (R75-12)) 3.111 67.1 94
## ((D74-7741) - (R75-12)) - ((N73-877) - Tracy) 197.111 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-882) - (R73-81)) 207.667 94.9 94
## ((D74-7741) - (R75-12)) - ((N73-882) - (R75-12)) 9.778 67.1 94
## ((D74-7741) - (R75-12)) - ((N73-882) - Tracy) 203.778 94.9 94
## ((D74-7741) - (R75-12)) - ((R73-81) - (R75-12)) 32.667 67.1 94
## ((D74-7741) - (R75-12)) - ((R73-81) - Tracy) 226.667 94.9 94
## ((D74-7741) - (R75-12)) - ((R75-12) - Tracy) 424.556 116.2 94
## ((D74-7741) - Tracy) - ((N72-137) - (N72-3058)) -116.667 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (N72-3148)) 118.667 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (N73-1102)) 54.111 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (N73-693)) -11.111 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (N73-877)) -44.000 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (N73-882)) -50.667 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (R73-81)) -73.556 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - (R75-12)) -271.444 94.9 94
## ((D74-7741) - Tracy) - ((N72-137) - Tracy) -77.444 67.1 94
## ((D74-7741) - Tracy) - ((N72-3058) - (N72-3148)) 271.889 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (N73-1102)) 207.333 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (N73-693)) 142.111 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (N73-877)) 109.222 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (N73-882)) 102.556 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (R73-81)) 79.667 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - (R75-12)) -118.222 94.9 94
## ((D74-7741) - Tracy) - ((N72-3058) - Tracy) 75.778 67.1 94
## ((D74-7741) - Tracy) - ((N72-3148) - (N73-1102)) -28.000 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - (N73-693)) -93.222 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - (N73-877)) -126.111 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - (N73-882)) -132.778 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - (R73-81)) -155.667 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - (R75-12)) -353.556 94.9 94
## ((D74-7741) - Tracy) - ((N72-3148) - Tracy) -159.556 67.1 94
## ((D74-7741) - Tracy) - ((N73-1102) - (N73-693)) -28.667 94.9 94
## ((D74-7741) - Tracy) - ((N73-1102) - (N73-877)) -61.556 94.9 94
## ((D74-7741) - Tracy) - ((N73-1102) - (N73-882)) -68.222 94.9 94
## ((D74-7741) - Tracy) - ((N73-1102) - (R73-81)) -91.111 94.9 94
## ((D74-7741) - Tracy) - ((N73-1102) - (R75-12)) -289.000 94.9 94
## ((D74-7741) - Tracy) - ((N73-1102) - Tracy) -95.000 67.1 94
## ((D74-7741) - Tracy) - ((N73-693) - (N73-877)) 3.667 94.9 94
## ((D74-7741) - Tracy) - ((N73-693) - (N73-882)) -3.000 94.9 94
## ((D74-7741) - Tracy) - ((N73-693) - (R73-81)) -25.889 94.9 94
## ((D74-7741) - Tracy) - ((N73-693) - (R75-12)) -223.778 94.9 94
## ((D74-7741) - Tracy) - ((N73-693) - Tracy) -29.778 67.1 94
## ((D74-7741) - Tracy) - ((N73-877) - (N73-882)) 29.889 94.9 94
## ((D74-7741) - Tracy) - ((N73-877) - (R73-81)) 7.000 94.9 94
## ((D74-7741) - Tracy) - ((N73-877) - (R75-12)) -190.889 94.9 94
## ((D74-7741) - Tracy) - ((N73-877) - Tracy) 3.111 67.1 94
## ((D74-7741) - Tracy) - ((N73-882) - (R73-81)) 13.667 94.9 94
## ((D74-7741) - Tracy) - ((N73-882) - (R75-12)) -184.222 94.9 94
## ((D74-7741) - Tracy) - ((N73-882) - Tracy) 9.778 67.1 94
## ((D74-7741) - Tracy) - ((R73-81) - (R75-12)) -161.333 94.9 94
## ((D74-7741) - Tracy) - ((R73-81) - Tracy) 32.667 67.1 94
## ((D74-7741) - Tracy) - ((R75-12) - Tracy) 230.556 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (N72-3148)) 235.333 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (N73-1102)) 170.778 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (N73-693)) 105.556 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (N73-877)) 72.667 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (N73-882)) 66.000 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (R73-81)) 43.111 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - (R75-12)) -154.778 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-137) - Tracy) 39.222 67.1 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (N72-3148)) 388.556 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-1102)) 324.000 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-693)) 258.778 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-877)) 225.889 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-882)) 219.222 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (R73-81)) 196.333 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - (R75-12)) -1.556 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3058) - Tracy) 192.444 116.2 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (N73-1102)) 88.667 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (N73-693)) 23.444 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (N73-877)) -9.444 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (N73-882)) -16.111 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (R73-81)) -39.000 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - (R75-12)) -236.889 94.9 94
## ((N72-137) - (N72-3058)) - ((N72-3148) - Tracy) -42.889 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - (N73-693)) 88.000 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - (N73-877)) 55.111 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - (N73-882)) 48.444 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - (R73-81)) 25.556 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - (R75-12)) -172.333 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-1102) - Tracy) 21.667 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-693) - (N73-877)) 120.333 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-693) - (N73-882)) 113.667 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-693) - (R73-81)) 90.778 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-693) - (R75-12)) -107.111 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-693) - Tracy) 86.889 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-877) - (N73-882)) 146.556 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-877) - (R73-81)) 123.667 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-877) - (R75-12)) -74.222 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-877) - Tracy) 119.778 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-882) - (R73-81)) 130.333 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-882) - (R75-12)) -67.556 94.9 94
## ((N72-137) - (N72-3058)) - ((N73-882) - Tracy) 126.444 94.9 94
## ((N72-137) - (N72-3058)) - ((R73-81) - (R75-12)) -44.667 94.9 94
## ((N72-137) - (N72-3058)) - ((R73-81) - Tracy) 149.333 94.9 94
## ((N72-137) - (N72-3058)) - ((R75-12) - Tracy) 347.222 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (N73-1102)) -64.556 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (N73-693)) -129.778 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (N73-877)) -162.667 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (N73-882)) -169.333 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (R73-81)) -192.222 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - (R75-12)) -390.111 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-137) - Tracy) -196.111 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (N72-3148)) 153.222 67.1 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (N73-1102)) 88.667 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (N73-693)) 23.444 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (N73-877)) -9.444 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (N73-882)) -16.111 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (R73-81)) -39.000 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - (R75-12)) -236.889 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3058) - Tracy) -42.889 94.9 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (N73-1102)) -146.667 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (N73-693)) -211.889 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (N73-877)) -244.778 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (N73-882)) -251.444 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (R73-81)) -274.333 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - (R75-12)) -472.222 116.2 94
## ((N72-137) - (N72-3148)) - ((N72-3148) - Tracy) -278.222 116.2 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - (N73-693)) -147.333 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - (N73-877)) -180.222 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - (N73-882)) -186.889 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - (R73-81)) -209.778 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - (R75-12)) -407.667 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-1102) - Tracy) -213.667 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-693) - (N73-877)) -115.000 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-693) - (N73-882)) -121.667 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-693) - (R73-81)) -144.556 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-693) - (R75-12)) -342.444 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-693) - Tracy) -148.444 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-877) - (N73-882)) -88.778 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-877) - (R73-81)) -111.667 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-877) - (R75-12)) -309.556 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-877) - Tracy) -115.556 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-882) - (R73-81)) -105.000 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-882) - (R75-12)) -302.889 94.9 94
## ((N72-137) - (N72-3148)) - ((N73-882) - Tracy) -108.889 94.9 94
## ((N72-137) - (N72-3148)) - ((R73-81) - (R75-12)) -280.000 94.9 94
## ((N72-137) - (N72-3148)) - ((R73-81) - Tracy) -86.000 94.9 94
## ((N72-137) - (N72-3148)) - ((R75-12) - Tracy) 111.889 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-137) - (N73-693)) -65.222 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-137) - (N73-877)) -98.111 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-137) - (N73-882)) -104.778 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-137) - (R73-81)) -127.667 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-137) - (R75-12)) -325.556 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-137) - Tracy) -131.556 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (N72-3148)) 217.778 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-1102)) 153.222 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-693)) 88.000 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-877)) 55.111 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-882)) 48.444 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (R73-81)) 25.556 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - (R75-12)) -172.333 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3058) - Tracy) 21.667 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (N73-1102)) -82.111 67.1 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (N73-693)) -147.333 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (N73-877)) -180.222 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (N73-882)) -186.889 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (R73-81)) -209.778 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - (R75-12)) -407.667 94.9 94
## ((N72-137) - (N73-1102)) - ((N72-3148) - Tracy) -213.667 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - (N73-693)) -82.778 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - (N73-877)) -115.667 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - (N73-882)) -122.333 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - (R73-81)) -145.222 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - (R75-12)) -343.111 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-1102) - Tracy) -149.111 116.2 94
## ((N72-137) - (N73-1102)) - ((N73-693) - (N73-877)) -50.444 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-693) - (N73-882)) -57.111 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-693) - (R73-81)) -80.000 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-693) - (R75-12)) -277.889 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-693) - Tracy) -83.889 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-877) - (N73-882)) -24.222 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-877) - (R73-81)) -47.111 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-877) - (R75-12)) -245.000 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-877) - Tracy) -51.000 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-882) - (R73-81)) -40.444 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-882) - (R75-12)) -238.333 94.9 94
## ((N72-137) - (N73-1102)) - ((N73-882) - Tracy) -44.333 94.9 94
## ((N72-137) - (N73-1102)) - ((R73-81) - (R75-12)) -215.444 94.9 94
## ((N72-137) - (N73-1102)) - ((R73-81) - Tracy) -21.444 94.9 94
## ((N72-137) - (N73-1102)) - ((R75-12) - Tracy) 176.444 94.9 94
## ((N72-137) - (N73-693)) - ((N72-137) - (N73-877)) -32.889 67.1 94
## ((N72-137) - (N73-693)) - ((N72-137) - (N73-882)) -39.556 67.1 94
## ((N72-137) - (N73-693)) - ((N72-137) - (R73-81)) -62.444 67.1 94
## ((N72-137) - (N73-693)) - ((N72-137) - (R75-12)) -260.333 67.1 94
## ((N72-137) - (N73-693)) - ((N72-137) - Tracy) -66.333 67.1 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (N72-3148)) 283.000 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (N73-1102)) 218.444 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (N73-693)) 153.222 67.1 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (N73-877)) 120.333 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (N73-882)) 113.667 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (R73-81)) 90.778 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - (R75-12)) -107.111 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3058) - Tracy) 86.889 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (N73-1102)) -16.889 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (N73-693)) -82.111 67.1 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (N73-877)) -115.000 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (N73-882)) -121.667 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (R73-81)) -144.556 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - (R75-12)) -342.444 94.9 94
## ((N72-137) - (N73-693)) - ((N72-3148) - Tracy) -148.444 94.9 94
## ((N72-137) - (N73-693)) - ((N73-1102) - (N73-693)) -17.556 67.1 94
## ((N72-137) - (N73-693)) - ((N73-1102) - (N73-877)) -50.444 94.9 94
## ((N72-137) - (N73-693)) - ((N73-1102) - (N73-882)) -57.111 94.9 94
## ((N72-137) - (N73-693)) - ((N73-1102) - (R73-81)) -80.000 94.9 94
## ((N72-137) - (N73-693)) - ((N73-1102) - (R75-12)) -277.889 94.9 94
## ((N72-137) - (N73-693)) - ((N73-1102) - Tracy) -83.889 94.9 94
## ((N72-137) - (N73-693)) - ((N73-693) - (N73-877)) 14.778 116.2 94
## ((N72-137) - (N73-693)) - ((N73-693) - (N73-882)) 8.111 116.2 94
## ((N72-137) - (N73-693)) - ((N73-693) - (R73-81)) -14.778 116.2 94
## ((N72-137) - (N73-693)) - ((N73-693) - (R75-12)) -212.667 116.2 94
## ((N72-137) - (N73-693)) - ((N73-693) - Tracy) -18.667 116.2 94
## ((N72-137) - (N73-693)) - ((N73-877) - (N73-882)) 41.000 94.9 94
## ((N72-137) - (N73-693)) - ((N73-877) - (R73-81)) 18.111 94.9 94
## ((N72-137) - (N73-693)) - ((N73-877) - (R75-12)) -179.778 94.9 94
## ((N72-137) - (N73-693)) - ((N73-877) - Tracy) 14.222 94.9 94
## ((N72-137) - (N73-693)) - ((N73-882) - (R73-81)) 24.778 94.9 94
## ((N72-137) - (N73-693)) - ((N73-882) - (R75-12)) -173.111 94.9 94
## ((N72-137) - (N73-693)) - ((N73-882) - Tracy) 20.889 94.9 94
## ((N72-137) - (N73-693)) - ((R73-81) - (R75-12)) -150.222 94.9 94
## ((N72-137) - (N73-693)) - ((R73-81) - Tracy) 43.778 94.9 94
## ((N72-137) - (N73-693)) - ((R75-12) - Tracy) 241.667 94.9 94
## ((N72-137) - (N73-877)) - ((N72-137) - (N73-882)) -6.667 67.1 94
## ((N72-137) - (N73-877)) - ((N72-137) - (R73-81)) -29.556 67.1 94
## ((N72-137) - (N73-877)) - ((N72-137) - (R75-12)) -227.444 67.1 94
## ((N72-137) - (N73-877)) - ((N72-137) - Tracy) -33.444 67.1 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (N72-3148)) 315.889 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (N73-1102)) 251.333 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (N73-693)) 186.111 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (N73-877)) 153.222 67.1 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (N73-882)) 146.556 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (R73-81)) 123.667 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - (R75-12)) -74.222 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3058) - Tracy) 119.778 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (N73-1102)) 16.000 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (N73-693)) -49.222 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (N73-877)) -82.111 67.1 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (N73-882)) -88.778 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (R73-81)) -111.667 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - (R75-12)) -309.556 94.9 94
## ((N72-137) - (N73-877)) - ((N72-3148) - Tracy) -115.556 94.9 94
## ((N72-137) - (N73-877)) - ((N73-1102) - (N73-693)) 15.333 94.9 94
## ((N72-137) - (N73-877)) - ((N73-1102) - (N73-877)) -17.556 67.1 94
## ((N72-137) - (N73-877)) - ((N73-1102) - (N73-882)) -24.222 94.9 94
## ((N72-137) - (N73-877)) - ((N73-1102) - (R73-81)) -47.111 94.9 94
## ((N72-137) - (N73-877)) - ((N73-1102) - (R75-12)) -245.000 94.9 94
## ((N72-137) - (N73-877)) - ((N73-1102) - Tracy) -51.000 94.9 94
## ((N72-137) - (N73-877)) - ((N73-693) - (N73-877)) 47.667 67.1 94
## ((N72-137) - (N73-877)) - ((N73-693) - (N73-882)) 41.000 94.9 94
## ((N72-137) - (N73-877)) - ((N73-693) - (R73-81)) 18.111 94.9 94
## ((N72-137) - (N73-877)) - ((N73-693) - (R75-12)) -179.778 94.9 94
## ((N72-137) - (N73-877)) - ((N73-693) - Tracy) 14.222 94.9 94
## ((N72-137) - (N73-877)) - ((N73-877) - (N73-882)) 73.889 116.2 94
## ((N72-137) - (N73-877)) - ((N73-877) - (R73-81)) 51.000 116.2 94
## ((N72-137) - (N73-877)) - ((N73-877) - (R75-12)) -146.889 116.2 94
## ((N72-137) - (N73-877)) - ((N73-877) - Tracy) 47.111 116.2 94
## ((N72-137) - (N73-877)) - ((N73-882) - (R73-81)) 57.667 94.9 94
## ((N72-137) - (N73-877)) - ((N73-882) - (R75-12)) -140.222 94.9 94
## ((N72-137) - (N73-877)) - ((N73-882) - Tracy) 53.778 94.9 94
## ((N72-137) - (N73-877)) - ((R73-81) - (R75-12)) -117.333 94.9 94
## ((N72-137) - (N73-877)) - ((R73-81) - Tracy) 76.667 94.9 94
## ((N72-137) - (N73-877)) - ((R75-12) - Tracy) 274.556 94.9 94
## ((N72-137) - (N73-882)) - ((N72-137) - (R73-81)) -22.889 67.1 94
## ((N72-137) - (N73-882)) - ((N72-137) - (R75-12)) -220.778 67.1 94
## ((N72-137) - (N73-882)) - ((N72-137) - Tracy) -26.778 67.1 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (N72-3148)) 322.556 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (N73-1102)) 258.000 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (N73-693)) 192.778 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (N73-877)) 159.889 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (N73-882)) 153.222 67.1 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (R73-81)) 130.333 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - (R75-12)) -67.556 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3058) - Tracy) 126.444 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (N73-1102)) 22.667 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (N73-693)) -42.556 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (N73-877)) -75.444 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (N73-882)) -82.111 67.1 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (R73-81)) -105.000 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - (R75-12)) -302.889 94.9 94
## ((N72-137) - (N73-882)) - ((N72-3148) - Tracy) -108.889 94.9 94
## ((N72-137) - (N73-882)) - ((N73-1102) - (N73-693)) 22.000 94.9 94
## ((N72-137) - (N73-882)) - ((N73-1102) - (N73-877)) -10.889 94.9 94
## ((N72-137) - (N73-882)) - ((N73-1102) - (N73-882)) -17.556 67.1 94
## ((N72-137) - (N73-882)) - ((N73-1102) - (R73-81)) -40.444 94.9 94
## ((N72-137) - (N73-882)) - ((N73-1102) - (R75-12)) -238.333 94.9 94
## ((N72-137) - (N73-882)) - ((N73-1102) - Tracy) -44.333 94.9 94
## ((N72-137) - (N73-882)) - ((N73-693) - (N73-877)) 54.333 94.9 94
## ((N72-137) - (N73-882)) - ((N73-693) - (N73-882)) 47.667 67.1 94
## ((N72-137) - (N73-882)) - ((N73-693) - (R73-81)) 24.778 94.9 94
## ((N72-137) - (N73-882)) - ((N73-693) - (R75-12)) -173.111 94.9 94
## ((N72-137) - (N73-882)) - ((N73-693) - Tracy) 20.889 94.9 94
## ((N72-137) - (N73-882)) - ((N73-877) - (N73-882)) 80.556 67.1 94
## ((N72-137) - (N73-882)) - ((N73-877) - (R73-81)) 57.667 94.9 94
## ((N72-137) - (N73-882)) - ((N73-877) - (R75-12)) -140.222 94.9 94
## ((N72-137) - (N73-882)) - ((N73-877) - Tracy) 53.778 94.9 94
## ((N72-137) - (N73-882)) - ((N73-882) - (R73-81)) 64.333 116.2 94
## ((N72-137) - (N73-882)) - ((N73-882) - (R75-12)) -133.556 116.2 94
## ((N72-137) - (N73-882)) - ((N73-882) - Tracy) 60.444 116.2 94
## ((N72-137) - (N73-882)) - ((R73-81) - (R75-12)) -110.667 94.9 94
## ((N72-137) - (N73-882)) - ((R73-81) - Tracy) 83.333 94.9 94
## ((N72-137) - (N73-882)) - ((R75-12) - Tracy) 281.222 94.9 94
## ((N72-137) - (R73-81)) - ((N72-137) - (R75-12)) -197.889 67.1 94
## ((N72-137) - (R73-81)) - ((N72-137) - Tracy) -3.889 67.1 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (N72-3148)) 345.444 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (N73-1102)) 280.889 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (N73-693)) 215.667 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (N73-877)) 182.778 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (N73-882)) 176.111 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (R73-81)) 153.222 67.1 94
## ((N72-137) - (R73-81)) - ((N72-3058) - (R75-12)) -44.667 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3058) - Tracy) 149.333 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (N73-1102)) 45.556 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (N73-693)) -19.667 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (N73-877)) -52.556 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (N73-882)) -59.222 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (R73-81)) -82.111 67.1 94
## ((N72-137) - (R73-81)) - ((N72-3148) - (R75-12)) -280.000 94.9 94
## ((N72-137) - (R73-81)) - ((N72-3148) - Tracy) -86.000 94.9 94
## ((N72-137) - (R73-81)) - ((N73-1102) - (N73-693)) 44.889 94.9 94
## ((N72-137) - (R73-81)) - ((N73-1102) - (N73-877)) 12.000 94.9 94
## ((N72-137) - (R73-81)) - ((N73-1102) - (N73-882)) 5.333 94.9 94
## ((N72-137) - (R73-81)) - ((N73-1102) - (R73-81)) -17.556 67.1 94
## ((N72-137) - (R73-81)) - ((N73-1102) - (R75-12)) -215.444 94.9 94
## ((N72-137) - (R73-81)) - ((N73-1102) - Tracy) -21.444 94.9 94
## ((N72-137) - (R73-81)) - ((N73-693) - (N73-877)) 77.222 94.9 94
## ((N72-137) - (R73-81)) - ((N73-693) - (N73-882)) 70.556 94.9 94
## ((N72-137) - (R73-81)) - ((N73-693) - (R73-81)) 47.667 67.1 94
## ((N72-137) - (R73-81)) - ((N73-693) - (R75-12)) -150.222 94.9 94
## ((N72-137) - (R73-81)) - ((N73-693) - Tracy) 43.778 94.9 94
## ((N72-137) - (R73-81)) - ((N73-877) - (N73-882)) 103.444 94.9 94
## ((N72-137) - (R73-81)) - ((N73-877) - (R73-81)) 80.556 67.1 94
## ((N72-137) - (R73-81)) - ((N73-877) - (R75-12)) -117.333 94.9 94
## ((N72-137) - (R73-81)) - ((N73-877) - Tracy) 76.667 94.9 94
## ((N72-137) - (R73-81)) - ((N73-882) - (R73-81)) 87.222 67.1 94
## ((N72-137) - (R73-81)) - ((N73-882) - (R75-12)) -110.667 94.9 94
## ((N72-137) - (R73-81)) - ((N73-882) - Tracy) 83.333 94.9 94
## ((N72-137) - (R73-81)) - ((R73-81) - (R75-12)) -87.778 116.2 94
## ((N72-137) - (R73-81)) - ((R73-81) - Tracy) 106.222 116.2 94
## ((N72-137) - (R73-81)) - ((R75-12) - Tracy) 304.111 94.9 94
## ((N72-137) - (R75-12)) - ((N72-137) - Tracy) 194.000 67.1 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (N72-3148)) 543.333 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (N73-1102)) 478.778 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (N73-693)) 413.556 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (N73-877)) 380.667 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (N73-882)) 374.000 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (R73-81)) 351.111 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3058) - (R75-12)) 153.222 67.1 94
## ((N72-137) - (R75-12)) - ((N72-3058) - Tracy) 347.222 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (N73-1102)) 243.444 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (N73-693)) 178.222 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (N73-877)) 145.333 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (N73-882)) 138.667 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (R73-81)) 115.778 94.9 94
## ((N72-137) - (R75-12)) - ((N72-3148) - (R75-12)) -82.111 67.1 94
## ((N72-137) - (R75-12)) - ((N72-3148) - Tracy) 111.889 94.9 94
## ((N72-137) - (R75-12)) - ((N73-1102) - (N73-693)) 242.778 94.9 94
## ((N72-137) - (R75-12)) - ((N73-1102) - (N73-877)) 209.889 94.9 94
## ((N72-137) - (R75-12)) - ((N73-1102) - (N73-882)) 203.222 94.9 94
## ((N72-137) - (R75-12)) - ((N73-1102) - (R73-81)) 180.333 94.9 94
## ((N72-137) - (R75-12)) - ((N73-1102) - (R75-12)) -17.556 67.1 94
## ((N72-137) - (R75-12)) - ((N73-1102) - Tracy) 176.444 94.9 94
## ((N72-137) - (R75-12)) - ((N73-693) - (N73-877)) 275.111 94.9 94
## ((N72-137) - (R75-12)) - ((N73-693) - (N73-882)) 268.444 94.9 94
## ((N72-137) - (R75-12)) - ((N73-693) - (R73-81)) 245.556 94.9 94
## ((N72-137) - (R75-12)) - ((N73-693) - (R75-12)) 47.667 67.1 94
## ((N72-137) - (R75-12)) - ((N73-693) - Tracy) 241.667 94.9 94
## ((N72-137) - (R75-12)) - ((N73-877) - (N73-882)) 301.333 94.9 94
## ((N72-137) - (R75-12)) - ((N73-877) - (R73-81)) 278.444 94.9 94
## ((N72-137) - (R75-12)) - ((N73-877) - (R75-12)) 80.556 67.1 94
## ((N72-137) - (R75-12)) - ((N73-877) - Tracy) 274.556 94.9 94
## ((N72-137) - (R75-12)) - ((N73-882) - (R73-81)) 285.111 94.9 94
## ((N72-137) - (R75-12)) - ((N73-882) - (R75-12)) 87.222 67.1 94
## ((N72-137) - (R75-12)) - ((N73-882) - Tracy) 281.222 94.9 94
## ((N72-137) - (R75-12)) - ((R73-81) - (R75-12)) 110.111 67.1 94
## ((N72-137) - (R75-12)) - ((R73-81) - Tracy) 304.111 94.9 94
## ((N72-137) - (R75-12)) - ((R75-12) - Tracy) 502.000 116.2 94
## ((N72-137) - Tracy) - ((N72-3058) - (N72-3148)) 349.333 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (N73-1102)) 284.778 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (N73-693)) 219.556 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (N73-877)) 186.667 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (N73-882)) 180.000 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (R73-81)) 157.111 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - (R75-12)) -40.778 94.9 94
## ((N72-137) - Tracy) - ((N72-3058) - Tracy) 153.222 67.1 94
## ((N72-137) - Tracy) - ((N72-3148) - (N73-1102)) 49.444 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - (N73-693)) -15.778 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - (N73-877)) -48.667 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - (N73-882)) -55.333 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - (R73-81)) -78.222 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - (R75-12)) -276.111 94.9 94
## ((N72-137) - Tracy) - ((N72-3148) - Tracy) -82.111 67.1 94
## ((N72-137) - Tracy) - ((N73-1102) - (N73-693)) 48.778 94.9 94
## ((N72-137) - Tracy) - ((N73-1102) - (N73-877)) 15.889 94.9 94
## ((N72-137) - Tracy) - ((N73-1102) - (N73-882)) 9.222 94.9 94
## ((N72-137) - Tracy) - ((N73-1102) - (R73-81)) -13.667 94.9 94
## ((N72-137) - Tracy) - ((N73-1102) - (R75-12)) -211.556 94.9 94
## ((N72-137) - Tracy) - ((N73-1102) - Tracy) -17.556 67.1 94
## ((N72-137) - Tracy) - ((N73-693) - (N73-877)) 81.111 94.9 94
## ((N72-137) - Tracy) - ((N73-693) - (N73-882)) 74.444 94.9 94
## ((N72-137) - Tracy) - ((N73-693) - (R73-81)) 51.556 94.9 94
## ((N72-137) - Tracy) - ((N73-693) - (R75-12)) -146.333 94.9 94
## ((N72-137) - Tracy) - ((N73-693) - Tracy) 47.667 67.1 94
## ((N72-137) - Tracy) - ((N73-877) - (N73-882)) 107.333 94.9 94
## ((N72-137) - Tracy) - ((N73-877) - (R73-81)) 84.444 94.9 94
## ((N72-137) - Tracy) - ((N73-877) - (R75-12)) -113.444 94.9 94
## ((N72-137) - Tracy) - ((N73-877) - Tracy) 80.556 67.1 94
## ((N72-137) - Tracy) - ((N73-882) - (R73-81)) 91.111 94.9 94
## ((N72-137) - Tracy) - ((N73-882) - (R75-12)) -106.778 94.9 94
## ((N72-137) - Tracy) - ((N73-882) - Tracy) 87.222 67.1 94
## ((N72-137) - Tracy) - ((R73-81) - (R75-12)) -83.889 94.9 94
## ((N72-137) - Tracy) - ((R73-81) - Tracy) 110.111 67.1 94
## ((N72-137) - Tracy) - ((R75-12) - Tracy) 308.000 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-1102)) -64.556 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-693)) -129.778 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-877)) -162.667 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-882)) -169.333 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (R73-81)) -192.222 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - (R75-12)) -390.111 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3058) - Tracy) -196.111 67.1 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-1102)) -299.889 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-693)) -365.111 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-877)) -398.000 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-882)) -404.667 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (R73-81)) -427.556 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - (R75-12)) -625.444 116.2 94
## ((N72-3058) - (N72-3148)) - ((N72-3148) - Tracy) -431.444 116.2 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - (N73-693)) -300.556 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - (N73-877)) -333.444 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - (N73-882)) -340.111 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - (R73-81)) -363.000 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - (R75-12)) -560.889 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-1102) - Tracy) -366.889 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-693) - (N73-877)) -268.222 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-693) - (N73-882)) -274.889 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-693) - (R73-81)) -297.778 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-693) - (R75-12)) -495.667 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-693) - Tracy) -301.667 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-877) - (N73-882)) -242.000 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-877) - (R73-81)) -264.889 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-877) - (R75-12)) -462.778 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-877) - Tracy) -268.778 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-882) - (R73-81)) -258.222 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-882) - (R75-12)) -456.111 94.9 94
## ((N72-3058) - (N72-3148)) - ((N73-882) - Tracy) -262.111 94.9 94
## ((N72-3058) - (N72-3148)) - ((R73-81) - (R75-12)) -433.222 94.9 94
## ((N72-3058) - (N72-3148)) - ((R73-81) - Tracy) -239.222 94.9 94
## ((N72-3058) - (N72-3148)) - ((R75-12) - Tracy) -41.333 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - (N73-693)) -65.222 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - (N73-877)) -98.111 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - (N73-882)) -104.778 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - (R73-81)) -127.667 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - (R75-12)) -325.556 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3058) - Tracy) -131.556 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (N73-1102)) -235.333 67.1 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (N73-693)) -300.556 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (N73-877)) -333.444 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (N73-882)) -340.111 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (R73-81)) -363.000 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - (R75-12)) -560.889 94.9 94
## ((N72-3058) - (N73-1102)) - ((N72-3148) - Tracy) -366.889 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - (N73-693)) -236.000 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - (N73-877)) -268.889 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - (N73-882)) -275.556 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - (R73-81)) -298.444 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - (R75-12)) -496.333 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-1102) - Tracy) -302.333 116.2 94
## ((N72-3058) - (N73-1102)) - ((N73-693) - (N73-877)) -203.667 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-693) - (N73-882)) -210.333 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-693) - (R73-81)) -233.222 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-693) - (R75-12)) -431.111 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-693) - Tracy) -237.111 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-877) - (N73-882)) -177.444 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-877) - (R73-81)) -200.333 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-877) - (R75-12)) -398.222 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-877) - Tracy) -204.222 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-882) - (R73-81)) -193.667 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-882) - (R75-12)) -391.556 94.9 94
## ((N72-3058) - (N73-1102)) - ((N73-882) - Tracy) -197.556 94.9 94
## ((N72-3058) - (N73-1102)) - ((R73-81) - (R75-12)) -368.667 94.9 94
## ((N72-3058) - (N73-1102)) - ((R73-81) - Tracy) -174.667 94.9 94
## ((N72-3058) - (N73-1102)) - ((R75-12) - Tracy) 23.222 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3058) - (N73-877)) -32.889 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3058) - (N73-882)) -39.556 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3058) - (R73-81)) -62.444 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3058) - (R75-12)) -260.333 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3058) - Tracy) -66.333 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (N73-1102)) -170.111 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (N73-693)) -235.333 67.1 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (N73-877)) -268.222 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (N73-882)) -274.889 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (R73-81)) -297.778 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - (R75-12)) -495.667 94.9 94
## ((N72-3058) - (N73-693)) - ((N72-3148) - Tracy) -301.667 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - (N73-693)) -170.778 67.1 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - (N73-877)) -203.667 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - (N73-882)) -210.333 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - (R73-81)) -233.222 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - (R75-12)) -431.111 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-1102) - Tracy) -237.111 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-693) - (N73-877)) -138.444 116.2 94
## ((N72-3058) - (N73-693)) - ((N73-693) - (N73-882)) -145.111 116.2 94
## ((N72-3058) - (N73-693)) - ((N73-693) - (R73-81)) -168.000 116.2 94
## ((N72-3058) - (N73-693)) - ((N73-693) - (R75-12)) -365.889 116.2 94
## ((N72-3058) - (N73-693)) - ((N73-693) - Tracy) -171.889 116.2 94
## ((N72-3058) - (N73-693)) - ((N73-877) - (N73-882)) -112.222 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-877) - (R73-81)) -135.111 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-877) - (R75-12)) -333.000 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-877) - Tracy) -139.000 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-882) - (R73-81)) -128.444 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-882) - (R75-12)) -326.333 94.9 94
## ((N72-3058) - (N73-693)) - ((N73-882) - Tracy) -132.333 94.9 94
## ((N72-3058) - (N73-693)) - ((R73-81) - (R75-12)) -303.444 94.9 94
## ((N72-3058) - (N73-693)) - ((R73-81) - Tracy) -109.444 94.9 94
## ((N72-3058) - (N73-693)) - ((R75-12) - Tracy) 88.444 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3058) - (N73-882)) -6.667 67.1 94
## ((N72-3058) - (N73-877)) - ((N72-3058) - (R73-81)) -29.556 67.1 94
## ((N72-3058) - (N73-877)) - ((N72-3058) - (R75-12)) -227.444 67.1 94
## ((N72-3058) - (N73-877)) - ((N72-3058) - Tracy) -33.444 67.1 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (N73-1102)) -137.222 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (N73-693)) -202.444 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (N73-877)) -235.333 67.1 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (N73-882)) -242.000 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (R73-81)) -264.889 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - (R75-12)) -462.778 94.9 94
## ((N72-3058) - (N73-877)) - ((N72-3148) - Tracy) -268.778 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - (N73-693)) -137.889 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - (N73-877)) -170.778 67.1 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - (N73-882)) -177.444 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - (R73-81)) -200.333 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - (R75-12)) -398.222 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-1102) - Tracy) -204.222 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-693) - (N73-877)) -105.556 67.1 94
## ((N72-3058) - (N73-877)) - ((N73-693) - (N73-882)) -112.222 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-693) - (R73-81)) -135.111 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-693) - (R75-12)) -333.000 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-693) - Tracy) -139.000 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-877) - (N73-882)) -79.333 116.2 94
## ((N72-3058) - (N73-877)) - ((N73-877) - (R73-81)) -102.222 116.2 94
## ((N72-3058) - (N73-877)) - ((N73-877) - (R75-12)) -300.111 116.2 94
## ((N72-3058) - (N73-877)) - ((N73-877) - Tracy) -106.111 116.2 94
## ((N72-3058) - (N73-877)) - ((N73-882) - (R73-81)) -95.556 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-882) - (R75-12)) -293.444 94.9 94
## ((N72-3058) - (N73-877)) - ((N73-882) - Tracy) -99.444 94.9 94
## ((N72-3058) - (N73-877)) - ((R73-81) - (R75-12)) -270.556 94.9 94
## ((N72-3058) - (N73-877)) - ((R73-81) - Tracy) -76.556 94.9 94
## ((N72-3058) - (N73-877)) - ((R75-12) - Tracy) 121.333 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3058) - (R73-81)) -22.889 67.1 94
## ((N72-3058) - (N73-882)) - ((N72-3058) - (R75-12)) -220.778 67.1 94
## ((N72-3058) - (N73-882)) - ((N72-3058) - Tracy) -26.778 67.1 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (N73-1102)) -130.556 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (N73-693)) -195.778 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (N73-877)) -228.667 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (N73-882)) -235.333 67.1 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (R73-81)) -258.222 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - (R75-12)) -456.111 94.9 94
## ((N72-3058) - (N73-882)) - ((N72-3148) - Tracy) -262.111 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - (N73-693)) -131.222 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - (N73-877)) -164.111 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - (N73-882)) -170.778 67.1 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - (R73-81)) -193.667 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - (R75-12)) -391.556 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-1102) - Tracy) -197.556 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-693) - (N73-877)) -98.889 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-693) - (N73-882)) -105.556 67.1 94
## ((N72-3058) - (N73-882)) - ((N73-693) - (R73-81)) -128.444 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-693) - (R75-12)) -326.333 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-693) - Tracy) -132.333 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-877) - (N73-882)) -72.667 67.1 94
## ((N72-3058) - (N73-882)) - ((N73-877) - (R73-81)) -95.556 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-877) - (R75-12)) -293.444 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-877) - Tracy) -99.444 94.9 94
## ((N72-3058) - (N73-882)) - ((N73-882) - (R73-81)) -88.889 116.2 94
## ((N72-3058) - (N73-882)) - ((N73-882) - (R75-12)) -286.778 116.2 94
## ((N72-3058) - (N73-882)) - ((N73-882) - Tracy) -92.778 116.2 94
## ((N72-3058) - (N73-882)) - ((R73-81) - (R75-12)) -263.889 94.9 94
## ((N72-3058) - (N73-882)) - ((R73-81) - Tracy) -69.889 94.9 94
## ((N72-3058) - (N73-882)) - ((R75-12) - Tracy) 128.000 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3058) - (R75-12)) -197.889 67.1 94
## ((N72-3058) - (R73-81)) - ((N72-3058) - Tracy) -3.889 67.1 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (N73-1102)) -107.667 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (N73-693)) -172.889 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (N73-877)) -205.778 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (N73-882)) -212.444 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (R73-81)) -235.333 67.1 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - (R75-12)) -433.222 94.9 94
## ((N72-3058) - (R73-81)) - ((N72-3148) - Tracy) -239.222 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - (N73-693)) -108.333 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - (N73-877)) -141.222 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - (N73-882)) -147.889 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - (R73-81)) -170.778 67.1 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - (R75-12)) -368.667 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-1102) - Tracy) -174.667 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-693) - (N73-877)) -76.000 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-693) - (N73-882)) -82.667 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-693) - (R73-81)) -105.556 67.1 94
## ((N72-3058) - (R73-81)) - ((N73-693) - (R75-12)) -303.444 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-693) - Tracy) -109.444 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-877) - (N73-882)) -49.778 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-877) - (R73-81)) -72.667 67.1 94
## ((N72-3058) - (R73-81)) - ((N73-877) - (R75-12)) -270.556 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-877) - Tracy) -76.556 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-882) - (R73-81)) -66.000 67.1 94
## ((N72-3058) - (R73-81)) - ((N73-882) - (R75-12)) -263.889 94.9 94
## ((N72-3058) - (R73-81)) - ((N73-882) - Tracy) -69.889 94.9 94
## ((N72-3058) - (R73-81)) - ((R73-81) - (R75-12)) -241.000 116.2 94
## ((N72-3058) - (R73-81)) - ((R73-81) - Tracy) -47.000 116.2 94
## ((N72-3058) - (R73-81)) - ((R75-12) - Tracy) 150.889 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3058) - Tracy) 194.000 67.1 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (N73-1102)) 90.222 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (N73-693)) 25.000 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (N73-877)) -7.889 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (N73-882)) -14.556 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (R73-81)) -37.444 94.9 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - (R75-12)) -235.333 67.1 94
## ((N72-3058) - (R75-12)) - ((N72-3148) - Tracy) -41.333 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - (N73-693)) 89.556 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - (N73-877)) 56.667 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - (N73-882)) 50.000 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - (R73-81)) 27.111 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - (R75-12)) -170.778 67.1 94
## ((N72-3058) - (R75-12)) - ((N73-1102) - Tracy) 23.222 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-693) - (N73-877)) 121.889 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-693) - (N73-882)) 115.222 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-693) - (R73-81)) 92.333 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-693) - (R75-12)) -105.556 67.1 94
## ((N72-3058) - (R75-12)) - ((N73-693) - Tracy) 88.444 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-877) - (N73-882)) 148.111 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-877) - (R73-81)) 125.222 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-877) - (R75-12)) -72.667 67.1 94
## ((N72-3058) - (R75-12)) - ((N73-877) - Tracy) 121.333 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-882) - (R73-81)) 131.889 94.9 94
## ((N72-3058) - (R75-12)) - ((N73-882) - (R75-12)) -66.000 67.1 94
## ((N72-3058) - (R75-12)) - ((N73-882) - Tracy) 128.000 94.9 94
## ((N72-3058) - (R75-12)) - ((R73-81) - (R75-12)) -43.111 67.1 94
## ((N72-3058) - (R75-12)) - ((R73-81) - Tracy) 150.889 94.9 94
## ((N72-3058) - (R75-12)) - ((R75-12) - Tracy) 348.778 116.2 94
## ((N72-3058) - Tracy) - ((N72-3148) - (N73-1102)) -103.778 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - (N73-693)) -169.000 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - (N73-877)) -201.889 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - (N73-882)) -208.556 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - (R73-81)) -231.444 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - (R75-12)) -429.333 94.9 94
## ((N72-3058) - Tracy) - ((N72-3148) - Tracy) -235.333 67.1 94
## ((N72-3058) - Tracy) - ((N73-1102) - (N73-693)) -104.444 94.9 94
## ((N72-3058) - Tracy) - ((N73-1102) - (N73-877)) -137.333 94.9 94
## ((N72-3058) - Tracy) - ((N73-1102) - (N73-882)) -144.000 94.9 94
## ((N72-3058) - Tracy) - ((N73-1102) - (R73-81)) -166.889 94.9 94
## ((N72-3058) - Tracy) - ((N73-1102) - (R75-12)) -364.778 94.9 94
## ((N72-3058) - Tracy) - ((N73-1102) - Tracy) -170.778 67.1 94
## ((N72-3058) - Tracy) - ((N73-693) - (N73-877)) -72.111 94.9 94
## ((N72-3058) - Tracy) - ((N73-693) - (N73-882)) -78.778 94.9 94
## ((N72-3058) - Tracy) - ((N73-693) - (R73-81)) -101.667 94.9 94
## ((N72-3058) - Tracy) - ((N73-693) - (R75-12)) -299.556 94.9 94
## ((N72-3058) - Tracy) - ((N73-693) - Tracy) -105.556 67.1 94
## ((N72-3058) - Tracy) - ((N73-877) - (N73-882)) -45.889 94.9 94
## ((N72-3058) - Tracy) - ((N73-877) - (R73-81)) -68.778 94.9 94
## ((N72-3058) - Tracy) - ((N73-877) - (R75-12)) -266.667 94.9 94
## ((N72-3058) - Tracy) - ((N73-877) - Tracy) -72.667 67.1 94
## ((N72-3058) - Tracy) - ((N73-882) - (R73-81)) -62.111 94.9 94
## ((N72-3058) - Tracy) - ((N73-882) - (R75-12)) -260.000 94.9 94
## ((N72-3058) - Tracy) - ((N73-882) - Tracy) -66.000 67.1 94
## ((N72-3058) - Tracy) - ((R73-81) - (R75-12)) -237.111 94.9 94
## ((N72-3058) - Tracy) - ((R73-81) - Tracy) -43.111 67.1 94
## ((N72-3058) - Tracy) - ((R75-12) - Tracy) 154.778 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-693)) -65.222 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-877)) -98.111 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-882)) -104.778 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - (R73-81)) -127.667 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - (R75-12)) -325.556 67.1 94
## ((N72-3148) - (N73-1102)) - ((N72-3148) - Tracy) -131.556 67.1 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-693)) -0.667 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-877)) -33.556 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-882)) -40.222 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - (R73-81)) -63.111 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - (R75-12)) -261.000 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-1102) - Tracy) -67.000 116.2 94
## ((N72-3148) - (N73-1102)) - ((N73-693) - (N73-877)) 31.667 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-693) - (N73-882)) 25.000 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-693) - (R73-81)) 2.111 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-693) - (R75-12)) -195.778 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-693) - Tracy) -1.778 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-877) - (N73-882)) 57.889 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-877) - (R73-81)) 35.000 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-877) - (R75-12)) -162.889 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-877) - Tracy) 31.111 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-882) - (R73-81)) 41.667 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-882) - (R75-12)) -156.222 94.9 94
## ((N72-3148) - (N73-1102)) - ((N73-882) - Tracy) 37.778 94.9 94
## ((N72-3148) - (N73-1102)) - ((R73-81) - (R75-12)) -133.333 94.9 94
## ((N72-3148) - (N73-1102)) - ((R73-81) - Tracy) 60.667 94.9 94
## ((N72-3148) - (N73-1102)) - ((R75-12) - Tracy) 258.556 94.9 94
## ((N72-3148) - (N73-693)) - ((N72-3148) - (N73-877)) -32.889 67.1 94
## ((N72-3148) - (N73-693)) - ((N72-3148) - (N73-882)) -39.556 67.1 94
## ((N72-3148) - (N73-693)) - ((N72-3148) - (R73-81)) -62.444 67.1 94
## ((N72-3148) - (N73-693)) - ((N72-3148) - (R75-12)) -260.333 67.1 94
## ((N72-3148) - (N73-693)) - ((N72-3148) - Tracy) -66.333 67.1 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-693)) 64.556 67.1 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-877)) 31.667 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-882)) 25.000 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - (R73-81)) 2.111 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - (R75-12)) -195.778 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-1102) - Tracy) -1.778 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-693) - (N73-877)) 96.889 116.2 94
## ((N72-3148) - (N73-693)) - ((N73-693) - (N73-882)) 90.222 116.2 94
## ((N72-3148) - (N73-693)) - ((N73-693) - (R73-81)) 67.333 116.2 94
## ((N72-3148) - (N73-693)) - ((N73-693) - (R75-12)) -130.556 116.2 94
## ((N72-3148) - (N73-693)) - ((N73-693) - Tracy) 63.444 116.2 94
## ((N72-3148) - (N73-693)) - ((N73-877) - (N73-882)) 123.111 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-877) - (R73-81)) 100.222 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-877) - (R75-12)) -97.667 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-877) - Tracy) 96.333 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-882) - (R73-81)) 106.889 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-882) - (R75-12)) -91.000 94.9 94
## ((N72-3148) - (N73-693)) - ((N73-882) - Tracy) 103.000 94.9 94
## ((N72-3148) - (N73-693)) - ((R73-81) - (R75-12)) -68.111 94.9 94
## ((N72-3148) - (N73-693)) - ((R73-81) - Tracy) 125.889 94.9 94
## ((N72-3148) - (N73-693)) - ((R75-12) - Tracy) 323.778 94.9 94
## ((N72-3148) - (N73-877)) - ((N72-3148) - (N73-882)) -6.667 67.1 94
## ((N72-3148) - (N73-877)) - ((N72-3148) - (R73-81)) -29.556 67.1 94
## ((N72-3148) - (N73-877)) - ((N72-3148) - (R75-12)) -227.444 67.1 94
## ((N72-3148) - (N73-877)) - ((N72-3148) - Tracy) -33.444 67.1 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-693)) 97.444 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-877)) 64.556 67.1 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-882)) 57.889 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - (R73-81)) 35.000 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - (R75-12)) -162.889 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-1102) - Tracy) 31.111 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-693) - (N73-877)) 129.778 67.1 94
## ((N72-3148) - (N73-877)) - ((N73-693) - (N73-882)) 123.111 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-693) - (R73-81)) 100.222 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-693) - (R75-12)) -97.667 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-693) - Tracy) 96.333 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-877) - (N73-882)) 156.000 116.2 94
## ((N72-3148) - (N73-877)) - ((N73-877) - (R73-81)) 133.111 116.2 94
## ((N72-3148) - (N73-877)) - ((N73-877) - (R75-12)) -64.778 116.2 94
## ((N72-3148) - (N73-877)) - ((N73-877) - Tracy) 129.222 116.2 94
## ((N72-3148) - (N73-877)) - ((N73-882) - (R73-81)) 139.778 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-882) - (R75-12)) -58.111 94.9 94
## ((N72-3148) - (N73-877)) - ((N73-882) - Tracy) 135.889 94.9 94
## ((N72-3148) - (N73-877)) - ((R73-81) - (R75-12)) -35.222 94.9 94
## ((N72-3148) - (N73-877)) - ((R73-81) - Tracy) 158.778 94.9 94
## ((N72-3148) - (N73-877)) - ((R75-12) - Tracy) 356.667 94.9 94
## ((N72-3148) - (N73-882)) - ((N72-3148) - (R73-81)) -22.889 67.1 94
## ((N72-3148) - (N73-882)) - ((N72-3148) - (R75-12)) -220.778 67.1 94
## ((N72-3148) - (N73-882)) - ((N72-3148) - Tracy) -26.778 67.1 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-693)) 104.111 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-877)) 71.222 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-882)) 64.556 67.1 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - (R73-81)) 41.667 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - (R75-12)) -156.222 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-1102) - Tracy) 37.778 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-693) - (N73-877)) 136.444 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-693) - (N73-882)) 129.778 67.1 94
## ((N72-3148) - (N73-882)) - ((N73-693) - (R73-81)) 106.889 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-693) - (R75-12)) -91.000 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-693) - Tracy) 103.000 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-877) - (N73-882)) 162.667 67.1 94
## ((N72-3148) - (N73-882)) - ((N73-877) - (R73-81)) 139.778 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-877) - (R75-12)) -58.111 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-877) - Tracy) 135.889 94.9 94
## ((N72-3148) - (N73-882)) - ((N73-882) - (R73-81)) 146.444 116.2 94
## ((N72-3148) - (N73-882)) - ((N73-882) - (R75-12)) -51.444 116.2 94
## ((N72-3148) - (N73-882)) - ((N73-882) - Tracy) 142.556 116.2 94
## ((N72-3148) - (N73-882)) - ((R73-81) - (R75-12)) -28.556 94.9 94
## ((N72-3148) - (N73-882)) - ((R73-81) - Tracy) 165.444 94.9 94
## ((N72-3148) - (N73-882)) - ((R75-12) - Tracy) 363.333 94.9 94
## ((N72-3148) - (R73-81)) - ((N72-3148) - (R75-12)) -197.889 67.1 94
## ((N72-3148) - (R73-81)) - ((N72-3148) - Tracy) -3.889 67.1 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-693)) 127.000 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-877)) 94.111 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-882)) 87.444 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - (R73-81)) 64.556 67.1 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - (R75-12)) -133.333 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-1102) - Tracy) 60.667 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-693) - (N73-877)) 159.333 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-693) - (N73-882)) 152.667 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-693) - (R73-81)) 129.778 67.1 94
## ((N72-3148) - (R73-81)) - ((N73-693) - (R75-12)) -68.111 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-693) - Tracy) 125.889 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-877) - (N73-882)) 185.556 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-877) - (R73-81)) 162.667 67.1 94
## ((N72-3148) - (R73-81)) - ((N73-877) - (R75-12)) -35.222 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-877) - Tracy) 158.778 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-882) - (R73-81)) 169.333 67.1 94
## ((N72-3148) - (R73-81)) - ((N73-882) - (R75-12)) -28.556 94.9 94
## ((N72-3148) - (R73-81)) - ((N73-882) - Tracy) 165.444 94.9 94
## ((N72-3148) - (R73-81)) - ((R73-81) - (R75-12)) -5.667 116.2 94
## ((N72-3148) - (R73-81)) - ((R73-81) - Tracy) 188.333 116.2 94
## ((N72-3148) - (R73-81)) - ((R75-12) - Tracy) 386.222 94.9 94
## ((N72-3148) - (R75-12)) - ((N72-3148) - Tracy) 194.000 67.1 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-693)) 324.889 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-877)) 292.000 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-882)) 285.333 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - (R73-81)) 262.444 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - (R75-12)) 64.556 67.1 94
## ((N72-3148) - (R75-12)) - ((N73-1102) - Tracy) 258.556 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-693) - (N73-877)) 357.222 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-693) - (N73-882)) 350.556 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-693) - (R73-81)) 327.667 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-693) - (R75-12)) 129.778 67.1 94
## ((N72-3148) - (R75-12)) - ((N73-693) - Tracy) 323.778 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-877) - (N73-882)) 383.444 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-877) - (R73-81)) 360.556 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-877) - (R75-12)) 162.667 67.1 94
## ((N72-3148) - (R75-12)) - ((N73-877) - Tracy) 356.667 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-882) - (R73-81)) 367.222 94.9 94
## ((N72-3148) - (R75-12)) - ((N73-882) - (R75-12)) 169.333 67.1 94
## ((N72-3148) - (R75-12)) - ((N73-882) - Tracy) 363.333 94.9 94
## ((N72-3148) - (R75-12)) - ((R73-81) - (R75-12)) 192.222 67.1 94
## ((N72-3148) - (R75-12)) - ((R73-81) - Tracy) 386.222 94.9 94
## ((N72-3148) - (R75-12)) - ((R75-12) - Tracy) 584.111 116.2 94
## ((N72-3148) - Tracy) - ((N73-1102) - (N73-693)) 130.889 94.9 94
## ((N72-3148) - Tracy) - ((N73-1102) - (N73-877)) 98.000 94.9 94
## ((N72-3148) - Tracy) - ((N73-1102) - (N73-882)) 91.333 94.9 94
## ((N72-3148) - Tracy) - ((N73-1102) - (R73-81)) 68.444 94.9 94
## ((N72-3148) - Tracy) - ((N73-1102) - (R75-12)) -129.444 94.9 94
## ((N72-3148) - Tracy) - ((N73-1102) - Tracy) 64.556 67.1 94
## ((N72-3148) - Tracy) - ((N73-693) - (N73-877)) 163.222 94.9 94
## ((N72-3148) - Tracy) - ((N73-693) - (N73-882)) 156.556 94.9 94
## ((N72-3148) - Tracy) - ((N73-693) - (R73-81)) 133.667 94.9 94
## ((N72-3148) - Tracy) - ((N73-693) - (R75-12)) -64.222 94.9 94
## ((N72-3148) - Tracy) - ((N73-693) - Tracy) 129.778 67.1 94
## ((N72-3148) - Tracy) - ((N73-877) - (N73-882)) 189.444 94.9 94
## ((N72-3148) - Tracy) - ((N73-877) - (R73-81)) 166.556 94.9 94
## ((N72-3148) - Tracy) - ((N73-877) - (R75-12)) -31.333 94.9 94
## ((N72-3148) - Tracy) - ((N73-877) - Tracy) 162.667 67.1 94
## ((N72-3148) - Tracy) - ((N73-882) - (R73-81)) 173.222 94.9 94
## ((N72-3148) - Tracy) - ((N73-882) - (R75-12)) -24.667 94.9 94
## ((N72-3148) - Tracy) - ((N73-882) - Tracy) 169.333 67.1 94
## ((N72-3148) - Tracy) - ((R73-81) - (R75-12)) -1.778 94.9 94
## ((N72-3148) - Tracy) - ((R73-81) - Tracy) 192.222 67.1 94
## ((N72-3148) - Tracy) - ((R75-12) - Tracy) 390.111 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-1102) - (N73-877)) -32.889 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-1102) - (N73-882)) -39.556 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-1102) - (R73-81)) -62.444 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-1102) - (R75-12)) -260.333 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-1102) - Tracy) -66.333 67.1 94
## ((N73-1102) - (N73-693)) - ((N73-693) - (N73-877)) 32.333 116.2 94
## ((N73-1102) - (N73-693)) - ((N73-693) - (N73-882)) 25.667 116.2 94
## ((N73-1102) - (N73-693)) - ((N73-693) - (R73-81)) 2.778 116.2 94
## ((N73-1102) - (N73-693)) - ((N73-693) - (R75-12)) -195.111 116.2 94
## ((N73-1102) - (N73-693)) - ((N73-693) - Tracy) -1.111 116.2 94
## ((N73-1102) - (N73-693)) - ((N73-877) - (N73-882)) 58.556 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-877) - (R73-81)) 35.667 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-877) - (R75-12)) -162.222 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-877) - Tracy) 31.778 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-882) - (R73-81)) 42.333 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-882) - (R75-12)) -155.556 94.9 94
## ((N73-1102) - (N73-693)) - ((N73-882) - Tracy) 38.444 94.9 94
## ((N73-1102) - (N73-693)) - ((R73-81) - (R75-12)) -132.667 94.9 94
## ((N73-1102) - (N73-693)) - ((R73-81) - Tracy) 61.333 94.9 94
## ((N73-1102) - (N73-693)) - ((R75-12) - Tracy) 259.222 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-1102) - (N73-882)) -6.667 67.1 94
## ((N73-1102) - (N73-877)) - ((N73-1102) - (R73-81)) -29.556 67.1 94
## ((N73-1102) - (N73-877)) - ((N73-1102) - (R75-12)) -227.444 67.1 94
## ((N73-1102) - (N73-877)) - ((N73-1102) - Tracy) -33.444 67.1 94
## ((N73-1102) - (N73-877)) - ((N73-693) - (N73-877)) 65.222 67.1 94
## ((N73-1102) - (N73-877)) - ((N73-693) - (N73-882)) 58.556 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-693) - (R73-81)) 35.667 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-693) - (R75-12)) -162.222 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-693) - Tracy) 31.778 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-877) - (N73-882)) 91.444 116.2 94
## ((N73-1102) - (N73-877)) - ((N73-877) - (R73-81)) 68.556 116.2 94
## ((N73-1102) - (N73-877)) - ((N73-877) - (R75-12)) -129.333 116.2 94
## ((N73-1102) - (N73-877)) - ((N73-877) - Tracy) 64.667 116.2 94
## ((N73-1102) - (N73-877)) - ((N73-882) - (R73-81)) 75.222 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-882) - (R75-12)) -122.667 94.9 94
## ((N73-1102) - (N73-877)) - ((N73-882) - Tracy) 71.333 94.9 94
## ((N73-1102) - (N73-877)) - ((R73-81) - (R75-12)) -99.778 94.9 94
## ((N73-1102) - (N73-877)) - ((R73-81) - Tracy) 94.222 94.9 94
## ((N73-1102) - (N73-877)) - ((R75-12) - Tracy) 292.111 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-1102) - (R73-81)) -22.889 67.1 94
## ((N73-1102) - (N73-882)) - ((N73-1102) - (R75-12)) -220.778 67.1 94
## ((N73-1102) - (N73-882)) - ((N73-1102) - Tracy) -26.778 67.1 94
## ((N73-1102) - (N73-882)) - ((N73-693) - (N73-877)) 71.889 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-693) - (N73-882)) 65.222 67.1 94
## ((N73-1102) - (N73-882)) - ((N73-693) - (R73-81)) 42.333 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-693) - (R75-12)) -155.556 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-693) - Tracy) 38.444 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-877) - (N73-882)) 98.111 67.1 94
## ((N73-1102) - (N73-882)) - ((N73-877) - (R73-81)) 75.222 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-877) - (R75-12)) -122.667 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-877) - Tracy) 71.333 94.9 94
## ((N73-1102) - (N73-882)) - ((N73-882) - (R73-81)) 81.889 116.2 94
## ((N73-1102) - (N73-882)) - ((N73-882) - (R75-12)) -116.000 116.2 94
## ((N73-1102) - (N73-882)) - ((N73-882) - Tracy) 78.000 116.2 94
## ((N73-1102) - (N73-882)) - ((R73-81) - (R75-12)) -93.111 94.9 94
## ((N73-1102) - (N73-882)) - ((R73-81) - Tracy) 100.889 94.9 94
## ((N73-1102) - (N73-882)) - ((R75-12) - Tracy) 298.778 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-1102) - (R75-12)) -197.889 67.1 94
## ((N73-1102) - (R73-81)) - ((N73-1102) - Tracy) -3.889 67.1 94
## ((N73-1102) - (R73-81)) - ((N73-693) - (N73-877)) 94.778 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-693) - (N73-882)) 88.111 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-693) - (R73-81)) 65.222 67.1 94
## ((N73-1102) - (R73-81)) - ((N73-693) - (R75-12)) -132.667 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-693) - Tracy) 61.333 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-877) - (N73-882)) 121.000 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-877) - (R73-81)) 98.111 67.1 94
## ((N73-1102) - (R73-81)) - ((N73-877) - (R75-12)) -99.778 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-877) - Tracy) 94.222 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-882) - (R73-81)) 104.778 67.1 94
## ((N73-1102) - (R73-81)) - ((N73-882) - (R75-12)) -93.111 94.9 94
## ((N73-1102) - (R73-81)) - ((N73-882) - Tracy) 100.889 94.9 94
## ((N73-1102) - (R73-81)) - ((R73-81) - (R75-12)) -70.222 116.2 94
## ((N73-1102) - (R73-81)) - ((R73-81) - Tracy) 123.778 116.2 94
## ((N73-1102) - (R73-81)) - ((R75-12) - Tracy) 321.667 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-1102) - Tracy) 194.000 67.1 94
## ((N73-1102) - (R75-12)) - ((N73-693) - (N73-877)) 292.667 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-693) - (N73-882)) 286.000 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-693) - (R73-81)) 263.111 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-693) - (R75-12)) 65.222 67.1 94
## ((N73-1102) - (R75-12)) - ((N73-693) - Tracy) 259.222 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-877) - (N73-882)) 318.889 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-877) - (R73-81)) 296.000 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-877) - (R75-12)) 98.111 67.1 94
## ((N73-1102) - (R75-12)) - ((N73-877) - Tracy) 292.111 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-882) - (R73-81)) 302.667 94.9 94
## ((N73-1102) - (R75-12)) - ((N73-882) - (R75-12)) 104.778 67.1 94
## ((N73-1102) - (R75-12)) - ((N73-882) - Tracy) 298.778 94.9 94
## ((N73-1102) - (R75-12)) - ((R73-81) - (R75-12)) 127.667 67.1 94
## ((N73-1102) - (R75-12)) - ((R73-81) - Tracy) 321.667 94.9 94
## ((N73-1102) - (R75-12)) - ((R75-12) - Tracy) 519.556 116.2 94
## ((N73-1102) - Tracy) - ((N73-693) - (N73-877)) 98.667 94.9 94
## ((N73-1102) - Tracy) - ((N73-693) - (N73-882)) 92.000 94.9 94
## ((N73-1102) - Tracy) - ((N73-693) - (R73-81)) 69.111 94.9 94
## ((N73-1102) - Tracy) - ((N73-693) - (R75-12)) -128.778 94.9 94
## ((N73-1102) - Tracy) - ((N73-693) - Tracy) 65.222 67.1 94
## ((N73-1102) - Tracy) - ((N73-877) - (N73-882)) 124.889 94.9 94
## ((N73-1102) - Tracy) - ((N73-877) - (R73-81)) 102.000 94.9 94
## ((N73-1102) - Tracy) - ((N73-877) - (R75-12)) -95.889 94.9 94
## ((N73-1102) - Tracy) - ((N73-877) - Tracy) 98.111 67.1 94
## ((N73-1102) - Tracy) - ((N73-882) - (R73-81)) 108.667 94.9 94
## ((N73-1102) - Tracy) - ((N73-882) - (R75-12)) -89.222 94.9 94
## ((N73-1102) - Tracy) - ((N73-882) - Tracy) 104.778 67.1 94
## ((N73-1102) - Tracy) - ((R73-81) - (R75-12)) -66.333 94.9 94
## ((N73-1102) - Tracy) - ((R73-81) - Tracy) 127.667 67.1 94
## ((N73-1102) - Tracy) - ((R75-12) - Tracy) 325.556 67.1 94
## ((N73-693) - (N73-877)) - ((N73-693) - (N73-882)) -6.667 67.1 94
## ((N73-693) - (N73-877)) - ((N73-693) - (R73-81)) -29.556 67.1 94
## ((N73-693) - (N73-877)) - ((N73-693) - (R75-12)) -227.444 67.1 94
## ((N73-693) - (N73-877)) - ((N73-693) - Tracy) -33.444 67.1 94
## ((N73-693) - (N73-877)) - ((N73-877) - (N73-882)) 26.222 116.2 94
## ((N73-693) - (N73-877)) - ((N73-877) - (R73-81)) 3.333 116.2 94
## ((N73-693) - (N73-877)) - ((N73-877) - (R75-12)) -194.556 116.2 94
## ((N73-693) - (N73-877)) - ((N73-877) - Tracy) -0.556 116.2 94
## ((N73-693) - (N73-877)) - ((N73-882) - (R73-81)) 10.000 94.9 94
## ((N73-693) - (N73-877)) - ((N73-882) - (R75-12)) -187.889 94.9 94
## ((N73-693) - (N73-877)) - ((N73-882) - Tracy) 6.111 94.9 94
## ((N73-693) - (N73-877)) - ((R73-81) - (R75-12)) -165.000 94.9 94
## ((N73-693) - (N73-877)) - ((R73-81) - Tracy) 29.000 94.9 94
## ((N73-693) - (N73-877)) - ((R75-12) - Tracy) 226.889 94.9 94
## ((N73-693) - (N73-882)) - ((N73-693) - (R73-81)) -22.889 67.1 94
## ((N73-693) - (N73-882)) - ((N73-693) - (R75-12)) -220.778 67.1 94
## ((N73-693) - (N73-882)) - ((N73-693) - Tracy) -26.778 67.1 94
## ((N73-693) - (N73-882)) - ((N73-877) - (N73-882)) 32.889 67.1 94
## ((N73-693) - (N73-882)) - ((N73-877) - (R73-81)) 10.000 94.9 94
## ((N73-693) - (N73-882)) - ((N73-877) - (R75-12)) -187.889 94.9 94
## ((N73-693) - (N73-882)) - ((N73-877) - Tracy) 6.111 94.9 94
## ((N73-693) - (N73-882)) - ((N73-882) - (R73-81)) 16.667 116.2 94
## ((N73-693) - (N73-882)) - ((N73-882) - (R75-12)) -181.222 116.2 94
## ((N73-693) - (N73-882)) - ((N73-882) - Tracy) 12.778 116.2 94
## ((N73-693) - (N73-882)) - ((R73-81) - (R75-12)) -158.333 94.9 94
## ((N73-693) - (N73-882)) - ((R73-81) - Tracy) 35.667 94.9 94
## ((N73-693) - (N73-882)) - ((R75-12) - Tracy) 233.556 94.9 94
## ((N73-693) - (R73-81)) - ((N73-693) - (R75-12)) -197.889 67.1 94
## ((N73-693) - (R73-81)) - ((N73-693) - Tracy) -3.889 67.1 94
## ((N73-693) - (R73-81)) - ((N73-877) - (N73-882)) 55.778 94.9 94
## ((N73-693) - (R73-81)) - ((N73-877) - (R73-81)) 32.889 67.1 94
## ((N73-693) - (R73-81)) - ((N73-877) - (R75-12)) -165.000 94.9 94
## ((N73-693) - (R73-81)) - ((N73-877) - Tracy) 29.000 94.9 94
## ((N73-693) - (R73-81)) - ((N73-882) - (R73-81)) 39.556 67.1 94
## ((N73-693) - (R73-81)) - ((N73-882) - (R75-12)) -158.333 94.9 94
## ((N73-693) - (R73-81)) - ((N73-882) - Tracy) 35.667 94.9 94
## ((N73-693) - (R73-81)) - ((R73-81) - (R75-12)) -135.444 116.2 94
## ((N73-693) - (R73-81)) - ((R73-81) - Tracy) 58.556 116.2 94
## ((N73-693) - (R73-81)) - ((R75-12) - Tracy) 256.444 94.9 94
## ((N73-693) - (R75-12)) - ((N73-693) - Tracy) 194.000 67.1 94
## ((N73-693) - (R75-12)) - ((N73-877) - (N73-882)) 253.667 94.9 94
## ((N73-693) - (R75-12)) - ((N73-877) - (R73-81)) 230.778 94.9 94
## ((N73-693) - (R75-12)) - ((N73-877) - (R75-12)) 32.889 67.1 94
## ((N73-693) - (R75-12)) - ((N73-877) - Tracy) 226.889 94.9 94
## ((N73-693) - (R75-12)) - ((N73-882) - (R73-81)) 237.444 94.9 94
## ((N73-693) - (R75-12)) - ((N73-882) - (R75-12)) 39.556 67.1 94
## ((N73-693) - (R75-12)) - ((N73-882) - Tracy) 233.556 94.9 94
## ((N73-693) - (R75-12)) - ((R73-81) - (R75-12)) 62.444 67.1 94
## ((N73-693) - (R75-12)) - ((R73-81) - Tracy) 256.444 94.9 94
## ((N73-693) - (R75-12)) - ((R75-12) - Tracy) 454.333 116.2 94
## ((N73-693) - Tracy) - ((N73-877) - (N73-882)) 59.667 94.9 94
## ((N73-693) - Tracy) - ((N73-877) - (R73-81)) 36.778 94.9 94
## ((N73-693) - Tracy) - ((N73-877) - (R75-12)) -161.111 94.9 94
## ((N73-693) - Tracy) - ((N73-877) - Tracy) 32.889 67.1 94
## ((N73-693) - Tracy) - ((N73-882) - (R73-81)) 43.444 94.9 94
## ((N73-693) - Tracy) - ((N73-882) - (R75-12)) -154.444 94.9 94
## ((N73-693) - Tracy) - ((N73-882) - Tracy) 39.556 67.1 94
## ((N73-693) - Tracy) - ((R73-81) - (R75-12)) -131.556 94.9 94
## ((N73-693) - Tracy) - ((R73-81) - Tracy) 62.444 67.1 94
## ((N73-693) - Tracy) - ((R75-12) - Tracy) 260.333 67.1 94
## ((N73-877) - (N73-882)) - ((N73-877) - (R73-81)) -22.889 67.1 94
## ((N73-877) - (N73-882)) - ((N73-877) - (R75-12)) -220.778 67.1 94
## ((N73-877) - (N73-882)) - ((N73-877) - Tracy) -26.778 67.1 94
## ((N73-877) - (N73-882)) - ((N73-882) - (R73-81)) -16.222 116.2 94
## ((N73-877) - (N73-882)) - ((N73-882) - (R75-12)) -214.111 116.2 94
## ((N73-877) - (N73-882)) - ((N73-882) - Tracy) -20.111 116.2 94
## ((N73-877) - (N73-882)) - ((R73-81) - (R75-12)) -191.222 94.9 94
## ((N73-877) - (N73-882)) - ((R73-81) - Tracy) 2.778 94.9 94
## ((N73-877) - (N73-882)) - ((R75-12) - Tracy) 200.667 94.9 94
## ((N73-877) - (R73-81)) - ((N73-877) - (R75-12)) -197.889 67.1 94
## ((N73-877) - (R73-81)) - ((N73-877) - Tracy) -3.889 67.1 94
## ((N73-877) - (R73-81)) - ((N73-882) - (R73-81)) 6.667 67.1 94
## ((N73-877) - (R73-81)) - ((N73-882) - (R75-12)) -191.222 94.9 94
## ((N73-877) - (R73-81)) - ((N73-882) - Tracy) 2.778 94.9 94
## ((N73-877) - (R73-81)) - ((R73-81) - (R75-12)) -168.333 116.2 94
## ((N73-877) - (R73-81)) - ((R73-81) - Tracy) 25.667 116.2 94
## ((N73-877) - (R73-81)) - ((R75-12) - Tracy) 223.556 94.9 94
## ((N73-877) - (R75-12)) - ((N73-877) - Tracy) 194.000 67.1 94
## ((N73-877) - (R75-12)) - ((N73-882) - (R73-81)) 204.556 94.9 94
## ((N73-877) - (R75-12)) - ((N73-882) - (R75-12)) 6.667 67.1 94
## ((N73-877) - (R75-12)) - ((N73-882) - Tracy) 200.667 94.9 94
## ((N73-877) - (R75-12)) - ((R73-81) - (R75-12)) 29.556 67.1 94
## ((N73-877) - (R75-12)) - ((R73-81) - Tracy) 223.556 94.9 94
## ((N73-877) - (R75-12)) - ((R75-12) - Tracy) 421.444 116.2 94
## ((N73-877) - Tracy) - ((N73-882) - (R73-81)) 10.556 94.9 94
## ((N73-877) - Tracy) - ((N73-882) - (R75-12)) -187.333 94.9 94
## ((N73-877) - Tracy) - ((N73-882) - Tracy) 6.667 67.1 94
## ((N73-877) - Tracy) - ((R73-81) - (R75-12)) -164.444 94.9 94
## ((N73-877) - Tracy) - ((R73-81) - Tracy) 29.556 67.1 94
## ((N73-877) - Tracy) - ((R75-12) - Tracy) 227.444 67.1 94
## ((N73-882) - (R73-81)) - ((N73-882) - (R75-12)) -197.889 67.1 94
## ((N73-882) - (R73-81)) - ((N73-882) - Tracy) -3.889 67.1 94
## ((N73-882) - (R73-81)) - ((R73-81) - (R75-12)) -175.000 116.2 94
## ((N73-882) - (R73-81)) - ((R73-81) - Tracy) 19.000 116.2 94
## ((N73-882) - (R73-81)) - ((R75-12) - Tracy) 216.889 94.9 94
## ((N73-882) - (R75-12)) - ((N73-882) - Tracy) 194.000 67.1 94
## ((N73-882) - (R75-12)) - ((R73-81) - (R75-12)) 22.889 67.1 94
## ((N73-882) - (R75-12)) - ((R73-81) - Tracy) 216.889 94.9 94
## ((N73-882) - (R75-12)) - ((R75-12) - Tracy) 414.778 116.2 94
## ((N73-882) - Tracy) - ((R73-81) - (R75-12)) -171.111 94.9 94
## ((N73-882) - Tracy) - ((R73-81) - Tracy) 22.889 67.1 94
## ((N73-882) - Tracy) - ((R75-12) - Tracy) 220.778 67.1 94
## ((R73-81) - (R75-12)) - ((R73-81) - Tracy) 194.000 67.1 94
## ((R73-81) - (R75-12)) - ((R75-12) - Tracy) 391.889 116.2 94
## ((R73-81) - Tracy) - ((R75-12) - Tracy) 197.889 67.1 94
## t.ratio p.value
## 1.155 1.0000
## -1.130 1.0000
## 2.379 0.9869
## 1.416 1.0000
## 0.444 1.0000
## -0.046 1.0000
## -0.146 1.0000
## -0.487 1.0000
## -3.437 0.4057
## -0.545 1.0000
## 0.566 1.0000
## -0.753 1.0000
## 1.273 1.0000
## 0.717 1.0000
## 0.156 1.0000
## -0.127 1.0000
## -0.185 1.0000
## -0.382 1.0000
## -2.085 0.9991
## -0.415 1.0000
## -1.738 1.0000
## 0.743 1.0000
## 0.062 1.0000
## -0.626 1.0000
## -0.972 1.0000
## -1.043 1.0000
## -1.284 1.0000
## -3.370 0.4566
## -1.325 1.0000
## 2.358 0.9888
## 1.677 1.0000
## 0.990 1.0000
## 0.643 1.0000
## 0.573 1.0000
## 0.332 1.0000
## -1.755 1.0000
## 0.291 1.0000
## -0.804 1.0000
## -1.491 1.0000
## -1.838 1.0000
## -1.908 0.9999
## -2.150 0.9982
## -4.236 0.0559
## -2.191 0.9973
## -0.811 1.0000
## -1.157 1.0000
## -1.228 1.0000
## -1.469 1.0000
## -3.555 0.3232
## -1.510 1.0000
## -0.470 1.0000
## -0.540 1.0000
## -0.781 1.0000
## -2.868 0.8319
## -0.822 1.0000
## -0.193 1.0000
## -0.435 1.0000
## -2.521 0.9663
## -0.476 1.0000
## -0.364 1.0000
## -2.451 0.9784
## -0.405 1.0000
## -2.209 0.9968
## -0.164 1.0000
## 1.922 0.9999
## -2.284 0.9938
## 1.224 1.0000
## 0.262 1.0000
## -0.711 1.0000
## -1.201 1.0000
## -1.300 1.0000
## -1.642 1.0000
## -4.592 0.0176
## -1.700 1.0000
## -0.174 1.0000
## -1.738 1.0000
## 0.743 1.0000
## 0.062 1.0000
## -0.626 1.0000
## -0.972 1.0000
## -1.043 1.0000
## -1.284 1.0000
## -3.370 0.4566
## -1.325 1.0000
## -2.086 0.9991
## -0.060 1.0000
## -0.616 1.0000
## -1.177 1.0000
## -1.460 1.0000
## -1.518 1.0000
## -1.715 1.0000
## -3.418 0.4199
## -1.748 1.0000
## 1.542 1.0000
## 0.861 1.0000
## 0.173 1.0000
## -0.173 1.0000
## -0.244 1.0000
## -0.485 1.0000
## -2.571 0.9550
## -0.526 1.0000
## -1.620 1.0000
## -2.308 0.9924
## -2.654 0.9306
## -2.725 0.9040
## -2.966 0.7692
## -5.052 0.0033
## -3.007 0.7405
## -1.627 1.0000
## -1.974 0.9998
## -2.044 0.9994
## -2.285 0.9937
## -4.372 0.0365
## -2.326 0.9912
## -1.286 1.0000
## -1.356 1.0000
## -1.598 1.0000
## -3.684 0.2449
## -1.639 1.0000
## -1.010 1.0000
## -1.251 1.0000
## -3.337 0.4821
## -1.292 1.0000
## -1.181 1.0000
## -3.267 0.5380
## -1.222 1.0000
## -3.026 0.7269
## -0.980 1.0000
## 1.106 1.0000
## 3.509 0.3546
## 2.546 0.9609
## 1.574 1.0000
## 1.083 1.0000
## 0.984 1.0000
## 0.643 1.0000
## -2.308 0.9924
## 0.585 1.0000
## 1.492 1.0000
## -0.174 1.0000
## 2.358 0.9888
## 1.677 1.0000
## 0.990 1.0000
## 0.643 1.0000
## 0.573 1.0000
## 0.332 1.0000
## -1.755 1.0000
## 0.291 1.0000
## -1.329 1.0000
## 1.542 1.0000
## 0.861 1.0000
## 0.173 1.0000
## -0.173 1.0000
## -0.244 1.0000
## -0.485 1.0000
## -2.571 0.9550
## -0.526 1.0000
## 2.578 0.9534
## 2.022 0.9996
## 1.460 1.0000
## 1.177 1.0000
## 1.120 1.0000
## 0.923 1.0000
## -0.780 1.0000
## 0.889 1.0000
## -0.005 1.0000
## -0.692 1.0000
## -1.039 1.0000
## -1.109 1.0000
## -1.351 1.0000
## -3.437 0.4061
## -1.392 1.0000
## -0.012 1.0000
## -0.358 1.0000
## -0.429 1.0000
## -0.670 1.0000
## -2.756 0.8901
## -0.711 1.0000
## 0.329 1.0000
## 0.259 1.0000
## 0.018 1.0000
## -2.069 0.9992
## -0.023 1.0000
## 0.606 1.0000
## 0.364 1.0000
## -1.722 1.0000
## 0.323 1.0000
## 0.435 1.0000
## -1.652 1.0000
## 0.394 1.0000
## -1.410 1.0000
## 0.635 1.0000
## 2.721 0.9055
## -0.962 1.0000
## -1.935 0.9999
## -2.425 0.9818
## -2.525 0.9656
## -2.866 0.8328
## -5.816 0.0002
## -2.924 0.7972
## -0.989 1.0000
## -2.604 0.9463
## -0.174 1.0000
## -0.804 1.0000
## -1.491 1.0000
## -1.838 1.0000
## -1.908 0.9999
## -2.150 0.9982
## -4.236 0.0559
## -2.191 0.9973
## -3.421 0.4183
## -1.329 1.0000
## -1.620 1.0000
## -2.308 0.9924
## -2.654 0.9306
## -2.725 0.9040
## -2.966 0.7692
## -5.052 0.0033
## -3.007 0.7405
## 0.956 1.0000
## -0.005 1.0000
## -0.692 1.0000
## -1.039 1.0000
## -1.109 1.0000
## -1.351 1.0000
## -3.437 0.4061
## -1.392 1.0000
## -2.030 0.9995
## -2.591 0.9499
## -2.874 0.8280
## -2.932 0.7922
## -3.129 0.6486
## -4.832 0.0075
## -3.162 0.6221
## -2.493 0.9716
## -2.839 0.8479
## -2.910 0.8062
## -3.151 0.6308
## -5.237 0.0016
## -3.192 0.5981
## -2.152 0.9982
## -2.222 0.9964
## -2.463 0.9765
## -4.550 0.0203
## -2.504 0.9695
## -1.875 0.9999
## -2.117 0.9987
## -4.203 0.0617
## -2.158 0.9981
## -2.046 0.9994
## -4.133 0.0760
## -2.087 0.9991
## -3.891 0.1480
## -1.846 1.0000
## 0.240 1.0000
## -0.972 1.0000
## -1.463 1.0000
## -1.562 1.0000
## -1.903 0.9999
## -4.854 0.0069
## -1.961 0.9998
## -0.308 1.0000
## -1.923 0.9999
## 0.558 1.0000
## -0.174 1.0000
## -0.811 1.0000
## -1.157 1.0000
## -1.228 1.0000
## -1.469 1.0000
## -3.555 0.3232
## -1.510 1.0000
## -2.740 0.8974
## -0.259 1.0000
## -1.329 1.0000
## -1.627 1.0000
## -1.974 0.9998
## -2.044 0.9994
## -2.285 0.9937
## -4.372 0.0365
## -2.326 0.9912
## 1.356 1.0000
## 0.956 1.0000
## -0.012 1.0000
## -0.358 1.0000
## -0.429 1.0000
## -0.670 1.0000
## -2.756 0.8901
## -0.711 1.0000
## -2.553 0.9594
## -2.493 0.9716
## -2.839 0.8479
## -2.910 0.8062
## -3.151 0.6308
## -5.237 0.0016
## -3.192 0.5981
## -1.480 1.0000
## -1.763 1.0000
## -1.820 1.0000
## -2.017 0.9996
## -3.721 0.2253
## -2.051 0.9994
## -1.471 1.0000
## -1.542 1.0000
## -1.783 1.0000
## -3.869 0.1567
## -1.824 1.0000
## -1.195 1.0000
## -1.436 1.0000
## -3.522 0.3452
## -1.477 1.0000
## -1.366 1.0000
## -3.452 0.3949
## -1.407 1.0000
## -3.211 0.5831
## -1.166 1.0000
## 0.921 1.0000
## -0.490 1.0000
## -0.590 1.0000
## -0.931 1.0000
## -3.881 0.1519
## -0.989 1.0000
## 0.380 1.0000
## -1.236 1.0000
## 1.245 1.0000
## 0.565 1.0000
## -0.174 1.0000
## -0.470 1.0000
## -0.540 1.0000
## -0.781 1.0000
## -2.868 0.8319
## -0.822 1.0000
## -2.052 0.9994
## 0.429 1.0000
## -0.252 1.0000
## -1.329 1.0000
## -1.286 1.0000
## -1.356 1.0000
## -1.598 1.0000
## -3.684 0.2449
## -1.639 1.0000
## 2.044 0.9994
## 1.364 1.0000
## 0.956 1.0000
## 0.329 1.0000
## 0.259 1.0000
## 0.018 1.0000
## -2.069 0.9992
## -0.023 1.0000
## -1.118 1.0000
## -2.553 0.9594
## -2.152 0.9982
## -2.222 0.9964
## -2.463 0.9765
## -4.550 0.0203
## -2.504 0.9695
## -1.590 1.0000
## -1.471 1.0000
## -1.542 1.0000
## -1.783 1.0000
## -3.869 0.1567
## -1.824 1.0000
## -0.640 1.0000
## -0.697 1.0000
## -0.894 1.0000
## -2.598 0.9481
## -0.928 1.0000
## -0.507 1.0000
## -0.749 1.0000
## -2.835 0.8505
## -0.790 1.0000
## -0.678 1.0000
## -2.765 0.8863
## -0.719 1.0000
## -2.523 0.9659
## -0.478 1.0000
## 1.608 1.0000
## -0.099 1.0000
## -0.441 1.0000
## -3.391 0.4405
## -0.499 1.0000
## 0.726 1.0000
## -0.889 1.0000
## 1.592 1.0000
## 0.911 1.0000
## 0.224 1.0000
## -0.174 1.0000
## -0.193 1.0000
## -0.435 1.0000
## -2.521 0.9663
## -0.476 1.0000
## -1.706 1.0000
## 0.775 1.0000
## 0.095 1.0000
## -0.593 1.0000
## -1.329 1.0000
## -1.010 1.0000
## -1.251 1.0000
## -3.337 0.4821
## -1.292 1.0000
## 2.391 0.9857
## 1.710 1.0000
## 1.023 1.0000
## 0.956 1.0000
## 0.606 1.0000
## 0.364 1.0000
## -1.722 1.0000
## 0.323 1.0000
## -0.771 1.0000
## -1.458 1.0000
## -2.553 0.9594
## -1.875 0.9999
## -2.117 0.9987
## -4.203 0.0617
## -2.158 0.9981
## -0.778 1.0000
## -1.590 1.0000
## -1.195 1.0000
## -1.436 1.0000
## -3.522 0.3452
## -1.477 1.0000
## -0.618 1.0000
## -0.507 1.0000
## -0.749 1.0000
## -2.835 0.8505
## -0.790 1.0000
## -0.131 1.0000
## -0.328 1.0000
## -2.031 0.9995
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## -0.341 1.0000
## -3.292 0.5183
## -0.399 1.0000
## 0.490 1.0000
## 0.105 1.0000
## -1.981 0.9997
## 0.064 1.0000
## 0.143 1.0000
## -1.560 1.0000
## 0.110 1.0000
## -1.669 1.0000
## 0.376 1.0000
## 2.462 0.9766
## -2.950 0.7797
## -0.058 1.0000
## 0.588 1.0000
## 0.490 1.0000
## -1.740 1.0000
## 0.306 1.0000
## 0.590 1.0000
## -1.669 1.0000
## 0.376 1.0000
## -1.166 1.0000
## 0.504 1.0000
## 2.704 0.9126
## 2.892 0.8170
## 2.674 0.9237
## 2.433 0.9808
## 0.490 1.0000
## 2.392 0.9856
## 2.503 0.9697
## 0.590 1.0000
## 2.462 0.9766
## 0.931 1.0000
## 2.704 0.9126
## 3.911 0.1407
## 0.629 1.0000
## 0.388 1.0000
## -1.699 1.0000
## 0.490 1.0000
## 0.458 1.0000
## -1.628 1.0000
## 0.590 1.0000
## -1.387 1.0000
## 0.931 1.0000
## 3.881 0.1519
## -0.341 1.0000
## -3.292 0.5183
## -0.399 1.0000
## -0.140 1.0000
## -1.843 1.0000
## -0.173 1.0000
## -2.016 0.9996
## 0.029 1.0000
## 2.116 0.9987
## -2.950 0.7797
## -0.058 1.0000
## 0.099 1.0000
## -2.016 0.9996
## 0.029 1.0000
## -1.449 1.0000
## 0.221 1.0000
## 2.357 0.9889
## 2.892 0.8170
## 2.157 0.9981
## 0.099 1.0000
## 2.116 0.9987
## 0.441 1.0000
## 2.357 0.9889
## 3.628 0.2774
## 0.111 1.0000
## -1.975 0.9998
## 0.099 1.0000
## -1.734 1.0000
## 0.441 1.0000
## 3.391 0.4405
## -2.950 0.7797
## -0.058 1.0000
## -1.506 1.0000
## 0.164 1.0000
## 2.287 0.9936
## 2.892 0.8170
## 0.341 1.0000
## 2.287 0.9936
## 3.570 0.3133
## -1.804 1.0000
## 0.341 1.0000
## 3.292 0.5183
## 2.892 0.8170
## 3.373 0.4541
## 2.950 0.7797
##
## Results are averaged over the levels of: loc
## P value adjustment: tukey method for comparing a family of 66 estimates
library(agricolae)
HSD.test(soy.lm.n, "gen", group = TRUE, console = TRUE)
##
## Study: soy.lm.n ~ "gen"
##
## HSD Test for yield
##
## Mean Square Error: 20243.4
##
## gen, means
##
## yield std r Min Max
## Centennial 1394.778 200.4631 9 1180 1713
## D74-7741 1406.444 301.6074 9 1099 1851
## N72-137 1483.889 188.6528 9 1177 1723
## N72-3058 1330.667 241.7861 9 990 1647
## N72-3148 1566.000 221.5835 9 1303 1929
## N73-1102 1501.444 228.3808 9 1178 1894
## N73-693 1436.222 163.4134 9 1214 1642
## N73-877 1403.333 236.8444 9 1107 1775
## N73-882 1396.667 191.5548 9 1096 1673
## R73-81 1373.778 271.7930 9 1064 1800
## R75-12 1175.889 181.7852 9 911 1422
## Tracy 1369.889 272.3850 9 960 1841
##
## Alpha: 0.05 ; DF Error: 94
## Critical Value of Studentized Range: 4.740283
##
## Minimun Significant Difference: 224.8147
##
## Treatments with the same letter are not significantly different.
##
## yield groups
## N72-3148 1566.000 a
## N73-1102 1501.444 ab
## N72-137 1483.889 ab
## N73-693 1436.222 ab
## D74-7741 1406.444 ab
## N73-877 1403.333 ab
## N73-882 1396.667 abc
## Centennial 1394.778 abc
## R73-81 1373.778 abc
## Tracy 1369.889 abc
## N72-3058 1330.667 bc
## R75-12 1175.889 c
The only significant difference is between (N72-3058) and (N72-3148) (p-value = 0.0318).
coef(summary(soy.lm.n))
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1193.583333 51.22641 23.30015787 7.831812e-41
## genD74-7741 11.666667 67.07111 0.17394474 8.622829e-01
## genN72-137 89.111111 67.07111 1.32860650 1.871942e-01
## genN72-3058 -64.111111 67.07111 -0.95586777 3.415909e-01
## genN72-3148 171.222222 67.07111 2.55284616 1.229518e-02
## genN73-1102 106.666667 67.07111 1.59035192 1.151132e-01
## genN73-693 41.444444 67.07111 0.61791799 5.381236e-01
## genN73-877 8.555556 67.07111 0.12755948 8.987701e-01
## genN73-882 1.888889 67.07111 0.02816248 9.775923e-01
## genR73-81 -21.000000 67.07111 -0.31310053 7.548979e-01
## genR75-12 -218.888889 67.07111 -3.26353467 1.534889e-03
## genTracy -24.888889 67.07111 -0.37108212 7.114114e-01
## locClinton 415.305556 33.53555 12.38403729 1.842719e-21
## locPlymouth 188.277778 33.53555 5.61427361 1.991500e-07
coef(summary(soy.lm.n))['(Intercept)', 'Estimate'] + coef(summary(soy.lm.n))['locPlymouth', 'Estimate']
## [1] 1381.861
#or
pred.df = data.frame(gen = "Centennial", loc = "Plymouth")
predict(soy.lm.n, pred.df)
## 1
## 1381.861
The average yield for the genotype ‘Centennial’ (reference level) and the location ‘Plymouth’ is 1381.861 kg
summary(soy.lm.n)
##
## Call:
## lm(formula = yield ~ gen + loc, data = soy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -329.00 -99.49 -13.31 82.08 440.31
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1193.583 51.226 23.300 < 2e-16 ***
## genD74-7741 11.667 67.071 0.174 0.86228
## genN72-137 89.111 67.071 1.329 0.18719
## genN72-3058 -64.111 67.071 -0.956 0.34159
## genN72-3148 171.222 67.071 2.553 0.01230 *
## genN73-1102 106.667 67.071 1.590 0.11511
## genN73-693 41.444 67.071 0.618 0.53812
## genN73-877 8.556 67.071 0.128 0.89877
## genN73-882 1.889 67.071 0.028 0.97759
## genR73-81 -21.000 67.071 -0.313 0.75490
## genR75-12 -218.889 67.071 -3.264 0.00153 **
## genTracy -24.889 67.071 -0.371 0.71141
## locClinton 415.306 33.536 12.384 < 2e-16 ***
## locPlymouth 188.278 33.536 5.614 1.99e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 142.3 on 94 degrees of freedom
## Multiple R-squared: 0.6797, Adjusted R-squared: 0.6354
## F-statistic: 15.35 on 13 and 94 DF, p-value: < 2.2e-16
yield.var = var(soy$yield)
yield.var
## [1] 55528.94
r.squared = var(fitted(soy.lm.n))/yield.var
r.squared
## [1] 0.6797359
The variance of the yield variable is 55528.94 and 67.97% is explained by the model.
library(car)
qqPlot(resid(soy.lm.n))
## [1] 63 75
plot(soy.lm.n, which=1)
c.) . . . then name the tools you use, give p values for each effect and answer the question.
soy.lme = lme(yield~gen*loc, random=~1|block, data=soy)
anova(soy.lme, type='marginal')
summary(soy.lme)
## Linear mixed-effects model fit by REML
## Data: soy
## AIC BIC logLik
## 1029.339 1115.853 -476.6696
##
## Random effects:
## Formula: ~1 | block
## (Intercept) Residual
## StdDev: 0.01037198 137.9402
##
## Fixed effects: yield ~ gen * loc
## Value Std.Error DF t-value p-value
## (Intercept) 1200.6667 79.63979 70 15.076215 0.0000
## genD74-7741 -93.6667 112.62768 70 -0.831649 0.4084
## genN72-137 249.6667 112.62768 70 2.216743 0.0299
## genN72-3058 -94.0000 112.62768 70 -0.834608 0.4068
## genN72-3148 127.0000 112.62768 70 1.127609 0.2633
## genN73-1102 62.0000 112.62768 70 0.550486 0.5837
## genN73-693 74.0000 112.62768 70 0.657032 0.5133
## genN73-877 12.0000 112.62768 70 0.106546 0.9155
## genN73-882 -26.6667 112.62768 70 -0.236768 0.8135
## genR73-81 -68.3333 112.62768 70 -0.606719 0.5460
## genR75-12 -100.3333 112.62768 70 -0.890841 0.3761
## genTracy -125.0000 112.62768 70 -1.109852 0.2709
## locClinton 391.0000 112.62768 70 3.471616 0.0009
## locPlymouth 191.3333 112.62768 70 1.698813 0.0938
## genD74-7741:locClinton 232.6667 159.27959 70 1.460744 0.1486
## genN72-137:locClinton -300.0000 159.27959 70 -1.883480 0.0638
## genN72-3058:locClinton 101.3333 159.27959 70 0.636198 0.5267
## genN72-3148:locClinton 65.3333 159.27959 70 0.410180 0.6829
## genN73-1102:locClinton 77.0000 159.27959 70 0.483427 0.6303
## genN73-693:locClinton -94.6667 159.27959 70 -0.594343 0.5542
## genN73-877:locClinton 15.6667 159.27959 70 0.098360 0.9219
## genN73-882:locClinton -41.6667 159.27959 70 -0.261595 0.7944
## genR73-81:locClinton 179.0000 159.27959 70 1.123810 0.2649
## genR75-12:locClinton -105.6667 159.27959 70 -0.663404 0.5093
## genTracy:locClinton 162.6667 159.27959 70 1.021265 0.3106
## genD74-7741:locPlymouth 83.3333 159.27959 70 0.523189 0.6025
## genN72-137:locPlymouth -181.6667 159.27959 70 -1.140552 0.2579
## genN72-3058:locPlymouth -11.6667 159.27959 70 -0.073246 0.9418
## genN72-3148:locPlymouth 67.3333 159.27959 70 0.422737 0.6738
## genN73-1102:locPlymouth 57.0000 159.27959 70 0.357861 0.7215
## genN73-693:locPlymouth -3.0000 159.27959 70 -0.018835 0.9850
## genN73-877:locPlymouth -26.0000 159.27959 70 -0.163235 0.8708
## genN73-882:locPlymouth 127.3333 159.27959 70 0.799433 0.4267
## genR73-81:locPlymouth -37.0000 159.27959 70 -0.232296 0.8170
## genR75-12:locPlymouth -250.0000 159.27959 70 -1.569567 0.1210
## genTracy:locPlymouth 137.6667 159.27959 70 0.864308 0.3904
## Correlation:
## (Intr) gnD74-7741 gnN72-137 gnN72-3058 gnN72-3148
## genD74-7741 -0.707
## genN72-137 -0.707 0.500
## genN72-3058 -0.707 0.500 0.500
## genN72-3148 -0.707 0.500 0.500 0.500
## genN73-1102 -0.707 0.500 0.500 0.500 0.500
## genN73-693 -0.707 0.500 0.500 0.500 0.500
## genN73-877 -0.707 0.500 0.500 0.500 0.500
## genN73-882 -0.707 0.500 0.500 0.500 0.500
## genR73-81 -0.707 0.500 0.500 0.500 0.500
## genR75-12 -0.707 0.500 0.500 0.500 0.500
## genTracy -0.707 0.500 0.500 0.500 0.500
## locClinton -0.707 0.500 0.500 0.500 0.500
## locPlymouth -0.707 0.500 0.500 0.500 0.500
## genD74-7741:locClinton 0.500 -0.707 -0.354 -0.354 -0.354
## genN72-137:locClinton 0.500 -0.354 -0.707 -0.354 -0.354
## genN72-3058:locClinton 0.500 -0.354 -0.354 -0.707 -0.354
## genN72-3148:locClinton 0.500 -0.354 -0.354 -0.354 -0.707
## genN73-1102:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-693:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-877:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-882:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genR73-81:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genR75-12:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genTracy:locClinton 0.500 -0.354 -0.354 -0.354 -0.354
## genD74-7741:locPlymouth 0.500 -0.707 -0.354 -0.354 -0.354
## genN72-137:locPlymouth 0.500 -0.354 -0.707 -0.354 -0.354
## genN72-3058:locPlymouth 0.500 -0.354 -0.354 -0.707 -0.354
## genN72-3148:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.707
## genN73-1102:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-693:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-877:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genN73-882:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genR73-81:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genR75-12:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## genTracy:locPlymouth 0.500 -0.354 -0.354 -0.354 -0.354
## gnN73-1102 gnN73-693 gnN73-877 gnN73-882 gnR73-81
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693 0.500
## genN73-877 0.500 0.500
## genN73-882 0.500 0.500 0.500
## genR73-81 0.500 0.500 0.500 0.500
## genR75-12 0.500 0.500 0.500 0.500 0.500
## genTracy 0.500 0.500 0.500 0.500 0.500
## locClinton 0.500 0.500 0.500 0.500 0.500
## locPlymouth 0.500 0.500 0.500 0.500 0.500
## genD74-7741:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-137:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-3058:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-3148:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genN73-1102:locClinton -0.707 -0.354 -0.354 -0.354 -0.354
## genN73-693:locClinton -0.354 -0.707 -0.354 -0.354 -0.354
## genN73-877:locClinton -0.354 -0.354 -0.707 -0.354 -0.354
## genN73-882:locClinton -0.354 -0.354 -0.354 -0.707 -0.354
## genR73-81:locClinton -0.354 -0.354 -0.354 -0.354 -0.707
## genR75-12:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genTracy:locClinton -0.354 -0.354 -0.354 -0.354 -0.354
## genD74-7741:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-137:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-3058:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## genN72-3148:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## genN73-1102:locPlymouth -0.707 -0.354 -0.354 -0.354 -0.354
## genN73-693:locPlymouth -0.354 -0.707 -0.354 -0.354 -0.354
## genN73-877:locPlymouth -0.354 -0.354 -0.707 -0.354 -0.354
## genN73-882:locPlymouth -0.354 -0.354 -0.354 -0.707 -0.354
## genR73-81:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.707
## genR75-12:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## genTracy:locPlymouth -0.354 -0.354 -0.354 -0.354 -0.354
## gnR75-12 gnTrcy lcClnt lcPlym gD74-7741:C gN72-137:C
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy 0.500
## locClinton 0.500 0.500
## locPlymouth 0.500 0.500 0.500
## genD74-7741:locClinton -0.354 -0.354 -0.707 -0.354
## genN72-137:locClinton -0.354 -0.354 -0.707 -0.354 0.500
## genN72-3058:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genN72-3148:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genN73-1102:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genN73-693:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genN73-877:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genN73-882:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genR73-81:locClinton -0.354 -0.354 -0.707 -0.354 0.500 0.500
## genR75-12:locClinton -0.707 -0.354 -0.707 -0.354 0.500 0.500
## genTracy:locClinton -0.354 -0.707 -0.707 -0.354 0.500 0.500
## genD74-7741:locPlymouth -0.354 -0.354 -0.354 -0.707 0.500 0.250
## genN72-137:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.500
## genN72-3058:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genN72-3148:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genN73-1102:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genN73-693:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genN73-877:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genN73-882:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genR73-81:locPlymouth -0.354 -0.354 -0.354 -0.707 0.250 0.250
## genR75-12:locPlymouth -0.707 -0.354 -0.354 -0.707 0.250 0.250
## genTracy:locPlymouth -0.354 -0.707 -0.354 -0.707 0.250 0.250
## gN72-3058:C gN72-3148:C gN73-1102:C gN73-693:C
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy
## locClinton
## locPlymouth
## genD74-7741:locClinton
## genN72-137:locClinton
## genN72-3058:locClinton
## genN72-3148:locClinton 0.500
## genN73-1102:locClinton 0.500 0.500
## genN73-693:locClinton 0.500 0.500 0.500
## genN73-877:locClinton 0.500 0.500 0.500 0.500
## genN73-882:locClinton 0.500 0.500 0.500 0.500
## genR73-81:locClinton 0.500 0.500 0.500 0.500
## genR75-12:locClinton 0.500 0.500 0.500 0.500
## genTracy:locClinton 0.500 0.500 0.500 0.500
## genD74-7741:locPlymouth 0.250 0.250 0.250 0.250
## genN72-137:locPlymouth 0.250 0.250 0.250 0.250
## genN72-3058:locPlymouth 0.500 0.250 0.250 0.250
## genN72-3148:locPlymouth 0.250 0.500 0.250 0.250
## genN73-1102:locPlymouth 0.250 0.250 0.500 0.250
## genN73-693:locPlymouth 0.250 0.250 0.250 0.500
## genN73-877:locPlymouth 0.250 0.250 0.250 0.250
## genN73-882:locPlymouth 0.250 0.250 0.250 0.250
## genR73-81:locPlymouth 0.250 0.250 0.250 0.250
## genR75-12:locPlymouth 0.250 0.250 0.250 0.250
## genTracy:locPlymouth 0.250 0.250 0.250 0.250
## gN73-877:C gN73-882:C gR73-81:C gR75-12:C gnTr:C
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy
## locClinton
## locPlymouth
## genD74-7741:locClinton
## genN72-137:locClinton
## genN72-3058:locClinton
## genN72-3148:locClinton
## genN73-1102:locClinton
## genN73-693:locClinton
## genN73-877:locClinton
## genN73-882:locClinton 0.500
## genR73-81:locClinton 0.500 0.500
## genR75-12:locClinton 0.500 0.500 0.500
## genTracy:locClinton 0.500 0.500 0.500 0.500
## genD74-7741:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN72-137:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN72-3058:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN72-3148:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN73-1102:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN73-693:locPlymouth 0.250 0.250 0.250 0.250 0.250
## genN73-877:locPlymouth 0.500 0.250 0.250 0.250 0.250
## genN73-882:locPlymouth 0.250 0.500 0.250 0.250 0.250
## genR73-81:locPlymouth 0.250 0.250 0.500 0.250 0.250
## genR75-12:locPlymouth 0.250 0.250 0.250 0.500 0.250
## genTracy:locPlymouth 0.250 0.250 0.250 0.250 0.500
## gD74-7741:P gN72-137:P gN72-3058:P gN72-3148:P
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy
## locClinton
## locPlymouth
## genD74-7741:locClinton
## genN72-137:locClinton
## genN72-3058:locClinton
## genN72-3148:locClinton
## genN73-1102:locClinton
## genN73-693:locClinton
## genN73-877:locClinton
## genN73-882:locClinton
## genR73-81:locClinton
## genR75-12:locClinton
## genTracy:locClinton
## genD74-7741:locPlymouth
## genN72-137:locPlymouth 0.500
## genN72-3058:locPlymouth 0.500 0.500
## genN72-3148:locPlymouth 0.500 0.500 0.500
## genN73-1102:locPlymouth 0.500 0.500 0.500 0.500
## genN73-693:locPlymouth 0.500 0.500 0.500 0.500
## genN73-877:locPlymouth 0.500 0.500 0.500 0.500
## genN73-882:locPlymouth 0.500 0.500 0.500 0.500
## genR73-81:locPlymouth 0.500 0.500 0.500 0.500
## genR75-12:locPlymouth 0.500 0.500 0.500 0.500
## genTracy:locPlymouth 0.500 0.500 0.500 0.500
## gN73-1102:P gN73-693:P gN73-877:P gN73-882:P gR73-81:P
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy
## locClinton
## locPlymouth
## genD74-7741:locClinton
## genN72-137:locClinton
## genN72-3058:locClinton
## genN72-3148:locClinton
## genN73-1102:locClinton
## genN73-693:locClinton
## genN73-877:locClinton
## genN73-882:locClinton
## genR73-81:locClinton
## genR75-12:locClinton
## genTracy:locClinton
## genD74-7741:locPlymouth
## genN72-137:locPlymouth
## genN72-3058:locPlymouth
## genN72-3148:locPlymouth
## genN73-1102:locPlymouth
## genN73-693:locPlymouth 0.500
## genN73-877:locPlymouth 0.500 0.500
## genN73-882:locPlymouth 0.500 0.500 0.500
## genR73-81:locPlymouth 0.500 0.500 0.500 0.500
## genR75-12:locPlymouth 0.500 0.500 0.500 0.500 0.500
## genTracy:locPlymouth 0.500 0.500 0.500 0.500 0.500
## gR75-12:P
## genD74-7741
## genN72-137
## genN72-3058
## genN72-3148
## genN73-1102
## genN73-693
## genN73-877
## genN73-882
## genR73-81
## genR75-12
## genTracy
## locClinton
## locPlymouth
## genD74-7741:locClinton
## genN72-137:locClinton
## genN72-3058:locClinton
## genN72-3148:locClinton
## genN73-1102:locClinton
## genN73-693:locClinton
## genN73-877:locClinton
## genN73-882:locClinton
## genR73-81:locClinton
## genR75-12:locClinton
## genTracy:locClinton
## genD74-7741:locPlymouth
## genN72-137:locPlymouth
## genN72-3058:locPlymouth
## genN72-3148:locPlymouth
## genN73-1102:locPlymouth
## genN73-693:locPlymouth
## genN73-877:locPlymouth
## genN73-882:locPlymouth
## genR73-81:locPlymouth
## genR75-12:locPlymouth
## genTracy:locPlymouth 0.500
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.98153540 -0.61983393 0.03020635 0.55700474 2.01536648
##
## Number of Observations: 108
## Number of Groups: 3
No significant interaction effect (p-value = 0.2211), we do find a significant genotype (p-value = 0.0462) and location (p-value = 0.0038) effect.
soy.lme.add = lme(yield~gen+loc, random=~1|block, data=soy)
library(emmeans)
soy.add.emm = emmeans(soy.lme.add, pairwise~gen)
summary(soy.add.emm)
## $emmeans
## gen emmean SE df lower.CL upper.CL
## Centennial 1395 47.4 2 1191 1599
## D74-7741 1406 47.4 2 1202 1611
## N72-137 1484 47.4 2 1280 1688
## N72-3058 1331 47.4 2 1127 1535
## N72-3148 1566 47.4 2 1362 1770
## N73-1102 1501 47.4 2 1297 1706
## N73-693 1436 47.4 2 1232 1640
## N73-877 1403 47.4 2 1199 1607
## N73-882 1397 47.4 2 1193 1601
## R73-81 1374 47.4 2 1170 1578
## R75-12 1176 47.4 2 972 1380
## Tracy 1370 47.4 2 1166 1574
##
## Results are averaged over the levels of: loc
## Degrees-of-freedom method: containment
## Confidence level used: 0.95
##
## $contrasts
## contrast estimate SE df t.ratio p.value
## Centennial - (D74-7741) -11.67 67.1 92 -0.174 1.0000
## Centennial - (N72-137) -89.11 67.1 92 -1.329 0.9731
## Centennial - (N72-3058) 64.11 67.1 92 0.956 0.9982
## Centennial - (N72-3148) -171.22 67.1 92 -2.553 0.3219
## Centennial - (N73-1102) -106.67 67.1 92 -1.590 0.9082
## Centennial - (N73-693) -41.44 67.1 92 -0.618 1.0000
## Centennial - (N73-877) -8.56 67.1 92 -0.128 1.0000
## Centennial - (N73-882) -1.89 67.1 92 -0.028 1.0000
## Centennial - (R73-81) 21.00 67.1 92 0.313 1.0000
## Centennial - (R75-12) 218.89 67.1 92 3.264 0.0640
## Centennial - Tracy 24.89 67.1 92 0.371 1.0000
## (D74-7741) - (N72-137) -77.44 67.1 92 -1.155 0.9910
## (D74-7741) - (N72-3058) 75.78 67.1 92 1.130 0.9925
## (D74-7741) - (N72-3148) -159.56 67.1 92 -2.379 0.4306
## (D74-7741) - (N73-1102) -95.00 67.1 92 -1.416 0.9573
## (D74-7741) - (N73-693) -29.78 67.1 92 -0.444 1.0000
## (D74-7741) - (N73-877) 3.11 67.1 92 0.046 1.0000
## (D74-7741) - (N73-882) 9.78 67.1 92 0.146 1.0000
## (D74-7741) - (R73-81) 32.67 67.1 92 0.487 1.0000
## (D74-7741) - (R75-12) 230.56 67.1 92 3.437 0.0394
## (D74-7741) - Tracy 36.56 67.1 92 0.545 1.0000
## (N72-137) - (N72-3058) 153.22 67.1 92 2.284 0.4949
## (N72-137) - (N72-3148) -82.11 67.1 92 -1.224 0.9856
## (N72-137) - (N73-1102) -17.56 67.1 92 -0.262 1.0000
## (N72-137) - (N73-693) 47.67 67.1 92 0.711 0.9999
## (N72-137) - (N73-877) 80.56 67.1 92 1.201 0.9876
## (N72-137) - (N73-882) 87.22 67.1 92 1.300 0.9771
## (N72-137) - (R73-81) 110.11 67.1 92 1.642 0.8887
## (N72-137) - (R75-12) 308.00 67.1 92 4.592 0.0008
## (N72-137) - Tracy 114.00 67.1 92 1.700 0.8638
## (N72-3058) - (N72-3148) -235.33 67.1 92 -3.509 0.0320
## (N72-3058) - (N73-1102) -170.78 67.1 92 -2.546 0.3257
## (N72-3058) - (N73-693) -105.56 67.1 92 -1.574 0.9140
## (N72-3058) - (N73-877) -72.67 67.1 92 -1.083 0.9947
## (N72-3058) - (N73-882) -66.00 67.1 92 -0.984 0.9977
## (N72-3058) - (R73-81) -43.11 67.1 92 -0.643 1.0000
## (N72-3058) - (R75-12) 154.78 67.1 92 2.308 0.4789
## (N72-3058) - Tracy -39.22 67.1 92 -0.585 1.0000
## (N72-3148) - (N73-1102) 64.56 67.1 92 0.962 0.9981
## (N72-3148) - (N73-693) 129.78 67.1 92 1.935 0.7344
## (N72-3148) - (N73-877) 162.67 67.1 92 2.425 0.4002
## (N72-3148) - (N73-882) 169.33 67.1 92 2.525 0.3384
## (N72-3148) - (R73-81) 192.22 67.1 92 2.866 0.1710
## (N72-3148) - (R75-12) 390.11 67.1 92 5.816 <.0001
## (N72-3148) - Tracy 196.11 67.1 92 2.924 0.1500
## (N73-1102) - (N73-693) 65.22 67.1 92 0.972 0.9979
## (N73-1102) - (N73-877) 98.11 67.1 92 1.463 0.9467
## (N73-1102) - (N73-882) 104.78 67.1 92 1.562 0.9180
## (N73-1102) - (R73-81) 127.67 67.1 92 1.903 0.7540
## (N73-1102) - (R75-12) 325.56 67.1 92 4.854 0.0003
## (N73-1102) - Tracy 131.56 67.1 92 1.961 0.7175
## (N73-693) - (N73-877) 32.89 67.1 92 0.490 1.0000
## (N73-693) - (N73-882) 39.56 67.1 92 0.590 1.0000
## (N73-693) - (R73-81) 62.44 67.1 92 0.931 0.9986
## (N73-693) - (R75-12) 260.33 67.1 92 3.881 0.0100
## (N73-693) - Tracy 66.33 67.1 92 0.989 0.9976
## (N73-877) - (N73-882) 6.67 67.1 92 0.099 1.0000
## (N73-877) - (R73-81) 29.56 67.1 92 0.441 1.0000
## (N73-877) - (R75-12) 227.44 67.1 92 3.391 0.0450
## (N73-877) - Tracy 33.44 67.1 92 0.499 1.0000
## (N73-882) - (R73-81) 22.89 67.1 92 0.341 1.0000
## (N73-882) - (R75-12) 220.78 67.1 92 3.292 0.0593
## (N73-882) - Tracy 26.78 67.1 92 0.399 1.0000
## (R73-81) - (R75-12) 197.89 67.1 92 2.950 0.1410
## (R73-81) - Tracy 3.89 67.1 92 0.058 1.0000
## (R75-12) - Tracy -194.00 67.1 92 -2.892 0.1611
##
## Results are averaged over the levels of: loc
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 12 estimates
The only significant difference is between genotype N72-3058 and N72-3148 (p-value = 0.0320). #### f.) According to the model from e.), how many kg of yield do we obtain for the Centennial genotype, grown in Plymouth, on average? Justify your answer.
fixef(summary(soy.lme.add))
## (Intercept) genD74-7741 genN72-137 genN72-3058 genN72-3148 genN73-1102
## 1193.583333 11.666667 89.111111 -64.111111 171.222222 106.666667
## genN73-693 genN73-877 genN73-882 genR73-81 genR75-12 genTracy
## 41.444444 8.555556 1.888889 -21.000000 -218.888889 -24.888889
## locClinton locPlymouth
## 415.305556 188.277778
fixef(summary(soy.lme.add))['(Intercept)'] + fixef(summary(soy.lme.add))['locPlymouth']
## (Intercept)
## 1381.861
The average yield of the Centennial genotype grown in Plymouth is 1381.861 kg.
yield.var = var(soy$yield)
library(MuMIn)
r.squaredGLMM(soy.lme.add)
## R2m R2c
## [1,] 0.6509061 0.6509061
library(agridat)
data('federer.tobacco')
str(federer.tobacco)
## 'data.frame': 56 obs. of 4 variables:
## $ row : int 1 1 1 1 1 1 1 1 2 2 ...
## $ block : int 1 2 3 4 5 6 7 8 1 2 ...
## $ dose : int 2500 250 0 2500 2500 5000 250 1000 5000 1500 ...
## $ height: num 1299 1369 1170 1219 1120 ...
federer.tobacco$dose = factor(federer.tobacco$dose)
federer.tobacco$row = factor(federer.tobacco$row)
federer.tobacco$block = factor(federer.tobacco$block)
library(nlme)
tob.lme = lme(height~dose, random=~1|block, data=federer.tobacco)
anova(tob.lme, type='marginal')
I performed an overall F test and we get that the dose effect is not significant (p-value = 0.1985)
The residuals are observed values − fitted values. Accordingly, the rows 2, 3, and 4 (with negative residuals) seem to produce systematically lower values than the rows 1,5, and 7.
tob.lme.row = lme(height~dose + row, random=~1|block, data=federer.tobacco)
anova(tob.lme.row, type='marginal')
When accounting for the fixed effect of the row we get a significant row effect (p-value=<.0001) and a significant dose effect (p-value=0.0281).
fixef(tob.lme.row)
## (Intercept) dose250 dose500 dose1000 dose1500 dose2500
## 1185.530716 23.070073 21.832451 2.666409 -32.124437 -6.580593
## dose5000 row2 row3 row4 row5 row6
## -128.810507 -243.365185 -400.603993 -251.300986 -47.647904 -129.270664
## row7
## -19.254680
fixef(tob.lme.row)['dose5000']
## dose5000
## -128.8105
The dose is fit as a reference level, we see that the average height decreases 128.8105 units for a typical block for a dose of 5000.
No it is not different for different rows since we do not take interaction effects between dose and row in consideration.
library(car)
qqPlot(resid(tob.lme.row))
## 1 2
## 49 10
plot(tob.lme.row)
We assess the normality assumption and the assumption of equal variance for the residuals, we do not check normality of the blocks since we only have 8 different blocks.
The qqPlot of the residuals does not show any problems with normality, the variance of the residuals seems to increase with the fitted values. This could be problematic.
We found problems with the variance, since we do not have problems with normality we do not want to transform the data but instead model heteroscedastic within group errors.
tob.lme.het = lme(height~dose+row, random=~1|block, data=federer.tobacco, weights=varIdent(form=~1|row))
plot(tob.lme.het)
library(emmeans)
tob.emm = emmeans(tob.lme.row, pairwise~dose)
tob.emm
## $emmeans
## dose emmean SE df lower.CL upper.CL
## 0 1030 43.8 7 926 1133
## 250 1053 43.2 7 951 1155
## 500 1051 44.6 7 946 1157
## 1000 1032 44.6 7 927 1138
## 1500 997 44.1 7 893 1102
## 2500 1023 43.6 7 920 1126
## 5000 901 42.7 7 800 1002
##
## Results are averaged over the levels of: row
## Degrees-of-freedom method: containment
## Confidence level used: 0.95
##
## $contrasts
## contrast estimate SE df t.ratio p.value
## 0 - 250 -23.07 44.1 36 -0.523 0.9983
## 0 - 500 -21.83 45.4 36 -0.481 0.9990
## 0 - 1000 -2.67 49.9 36 -0.053 1.0000
## 0 - 1500 32.12 46.9 36 0.685 0.9927
## 0 - 2500 6.58 47.2 36 0.140 1.0000
## 0 - 5000 128.81 45.5 36 2.831 0.0966
## 250 - 500 1.24 47.0 36 0.026 1.0000
## 250 - 1000 20.40 48.1 36 0.424 0.9995
## 250 - 1500 55.19 46.2 36 1.195 0.8913
## 250 - 2500 29.65 44.6 36 0.666 0.9937
## 250 - 5000 151.88 45.2 36 3.359 0.0280
## 500 - 1000 19.17 47.8 36 0.401 0.9996
## 500 - 1500 53.96 49.7 36 1.087 0.9280
## 500 - 2500 28.41 48.4 36 0.588 0.9968
## 500 - 5000 150.64 46.3 36 3.251 0.0366
## 1000 - 1500 34.79 47.0 36 0.740 0.9890
## 1000 - 2500 9.25 46.1 36 0.200 1.0000
## 1000 - 5000 131.48 45.7 36 2.874 0.0878
## 1500 - 2500 -25.54 46.8 36 -0.546 0.9979
## 1500 - 5000 96.69 44.5 36 2.175 0.3336
## 2500 - 5000 122.23 44.8 36 2.726 0.1207
##
## Results are averaged over the levels of: row
## Degrees-of-freedom method: containment
## P value adjustment: tukey method for comparing a family of 7 estimates
library(multcomp)
summary(glht(tob.lme.row, mcp(dose='Sequen')))
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Sequen Contrasts
##
##
## Fit: lme.formula(fixed = height ~ dose + row, data = federer.tobacco,
## random = ~1 | block)
##
## Linear Hypotheses:
## Estimate Std. Error z value Pr(>|z|)
## 250 - 0 == 0 23.070 44.140 0.523 0.9913
## 500 - 250 == 0 -1.238 47.047 -0.026 1.0000
## 1000 - 500 == 0 -19.166 47.811 -0.401 0.9978
## 1500 - 1000 == 0 -34.791 47.044 -0.740 0.9560
## 2500 - 1500 == 0 25.544 46.761 0.546 0.9892
## 5000 - 2500 == 0 -122.230 44.833 -2.726 0.0356 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
0 - 250 -23.07 44.1 36 -0.523 0.9983 250 - 500 1.24 47.0 36 0.026 1.0000 250 - 500 1.24 47.0 36 0.026 1.0000 1000 - 1500 34.79 47.0 36 0.740 0.9890 1500 - 2500 -25.54 46.8 36 -0.546 0.9979 2500 - 5000 122.23 44.8 36 2.726 0.1207
None of the comparisons are significant according to the emmeans method since it makes more comparisons and is thus stricter controling the familywise error rate at 5%. When comparing with the ‘Sequen’ method of the multcomp package we do get a significant difference for the difference of level 2500 and 5000 with a p-value of 0.0354. This method has a less strict p-value correction.
library(agridat)
data('gregory.cotton')
str(gregory.cotton)
## 'data.frame': 144 obs. of 6 variables:
## $ yield : num 0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
## $ year : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
## $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
## $ date : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ water : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
## $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...
cotton.Y1 = gregory.cotton[gregory.cotton$year == 'Y1' & gregory.cotton$nitrogen == 'N1', ]
with(cotton.Y1, tapply(yield, date, mean))
## D1 D2 D3 D4
## 2.773333 3.074444 2.664444 1.735556
The highest average yield is reported for date D2 with 3.074units.
cotton.lm = lm(yield~nitrogen*date, data=gregory.cotton)
library(car)
Anova(cotton.lm, type=2)
We perform an overall F test and get a significant interaction effect of nitrogen and date (p-value = 0.001748). We conclude that the effect of the nitrogen fertilizer does depend on the date.
unique(gregory.cotton$nitrogen)
## [1] N0 N1
## Levels: N0 N1
unique(gregory.cotton$date)
## [1] D1 D2 D3 D4
## Levels: D1 D2 D3 D4
We have two levels of nitrogen and four levels for dates, one of the levels each is the reference level, hence we get three interaction terms. (4-1)\(\cdot\)(2-1)=3
The interaction plot is more or less parallel for the interactions of D1, D2 and D3, for D4 the lines are very close together, hence we assume that the interaction between D4 and N1 is the most important term.
summary(cotton.lm)
##
## Call:
## lm(formula = yield ~ nitrogen * date, data = gregory.cotton)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9900 -0.3644 -0.1153 0.3664 1.2700
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.15389 0.11511 10.024 < 2e-16 ***
## nitrogenN1 1.14167 0.16279 7.013 9.93e-11 ***
## dateD2 0.33278 0.16279 2.044 0.042857 *
## dateD3 0.25722 0.16279 1.580 0.116401
## dateD4 -0.01944 0.16279 -0.119 0.905097
## nitrogenN1:dateD2 -0.04667 0.23021 -0.203 0.839665
## nitrogenN1:dateD3 -0.33278 0.23021 -1.446 0.150613
## nitrogenN1:dateD4 -0.81556 0.23021 -3.543 0.000543 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4884 on 136 degrees of freedom
## Multiple R-squared: 0.5502, Adjusted R-squared: 0.5271
## F-statistic: 23.77 on 7 and 136 DF, p-value: < 2.2e-16
The interaction terms for nitrogen level N1 and date level D2 as well as between nitrogen level N1 and date level D3 are not significant, with p-values 0.839665 and 0.150613 respectively. We do have a significant interaction term for nitrogen level N1 and date level D4 with p-value = 0.000543.
library(ggplot2)
ggplot(gregory.cotton, aes(date, yield)) + geom_point() + facet_grid(year~nitrogen)
cotton.lm.year = lm(yield~nitrogen*date*year, gregory.cotton)
library(car)
Anova(cotton.lm.year, type=2)
The nitrogen and year interaction effect is significant with a p-value of 0.02743. We conclude that the effect of the nitrogen on the yield depends on the year.
coef(summary(cotton.lm.year))
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.3177778 0.09949021 13.245301 9.397591e-26
## nitrogenN1 1.4555556 0.14070041 10.345070 1.333739e-18
## dateD2 0.5711111 0.14070041 4.059058 8.522107e-05
## dateD3 0.4800000 0.14070041 3.411504 8.650602e-04
## dateD4 0.1100000 0.14070041 0.781803 4.357741e-01
## yearY2 -0.3277778 0.14070041 -2.329615 2.139219e-02
## nitrogenN1:dateD2 -0.2700000 0.19898043 -1.356917 1.771956e-01
## nitrogenN1:dateD3 -0.5888889 0.19898043 -2.959532 3.671954e-03
## nitrogenN1:dateD4 -1.1477778 0.19898043 -5.768295 5.696978e-08
## nitrogenN1:yearY2 -0.6277778 0.19898043 -3.154973 2.000966e-03
## dateD2:yearY2 -0.4766667 0.19898043 -2.395546 1.804359e-02
## dateD3:yearY2 -0.4455556 0.19898043 -2.239193 2.686976e-02
## dateD4:yearY2 -0.2588889 0.19898043 -1.301077 1.955692e-01
## nitrogenN1:dateD2:yearY2 0.4466667 0.28140082 1.587297 1.149127e-01
## nitrogenN1:dateD3:yearY2 0.5122222 0.28140082 1.820258 7.105594e-02
## nitrogenN1:dateD4:yearY2 0.6644444 0.28140082 2.361203 1.972497e-02
coef(summary(cotton.lm.year))['(Intercept)', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1', 'Estimate'] + coef(summary(cotton.lm.year))['dateD4', 'Estimate'] + coef(summary(cotton.lm.year))['yearY2', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1:dateD4', 'Estimate'] + coef(summary(cotton.lm.year))['dateD4:yearY2', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1:dateD4:yearY2', 'Estimate']
## [1] 1.813333
#or
df.pred = data.frame(date = "D4", nitrogen = "N1", year = "Y2")
predict(cotton.lm.year, df.pred)
## 1
## 1.185556
The average yield is 1.813333.